ISE 514 project

engineering project and need the explanation and answer to help me learn.

HELLO,
I have a project that requires a full report and a PowerPoint. My problem is chapter 13.5 and you can find in the attached book. The name of the chapter is Applications and Extensions of Cyclic Staffing. I need quality work and NO AI. professor can detect plagiarism and AI be careful
Requirements: 2500
Final Project ISE 514 – Fall 2023 Due date: Nov. 21st, 2023, by 1:00 PM Each Presentation – 20 min. This assignment entails delving into an array of Advanced-level production and scheduling algorithms that are extensively used in the designated field to tackle real-world problems. The objective is to equip students with practical skills in applying scheduling algorithms to generate efficient solutions. Moreover, the project facilitates the comprehension of the operational protocols in different sectors. Make use of this task as a platform to broaden your expertise in the area of Advanced-level Production and Scheduling Algorithms. 1. In this project, you have been assigned a topic. 2. Your task is to study and comprehend the assigned topic thoroughly. 3. Next, you need to prepare 5 different quantitative problems/ examples based on the assigned topic and clearly explain notations, assumptions and complete procedure to solve each problem. 4. Provide a detailed explanation and clearly describe the steps involved in arriving at the solution. 5. In conclusion, clearly present the solution, and final results for each example. 6. Your final report and ppt must be prepared in your own words. I encourage you to put in your best effort and demonstrate your understanding of the concept. The project concerns should be discussed during professor’s office hours. Due date: – Nov. 21st, 2023, by 1:00 PM Submission Procedure – Bring the PROJECT REPORT printout and PPT printout to the class on Nov. 21st, 2023, submit your ppt and report printouts before the first
presentation and email the PROJECT REPORT and PPT to me (shalinig@usc.edu) by Nov. 21st, 2023 by 1:00 PM. The report should include the following sections. 1. Cover page a. Title of Topic b. Course name c. Due Date d. Team Members and ID 2. Organize the report into 6 sections i. Introduction ii. Clearly mention the Assumptions, Notations, Formulas, and Equations related to the assigned topic. iii. Methodology/Procedure / Detailed step-by-step procedure to solve each problem. iv. Only 5 different problems relevant to the assigned topic. Make sure to select a bit difficult problems. v. Result vi. List of References Grades will be based upon 1.General writing guidelines, including grammar, spelling, headers, footers, page numbering, slide numbers, diagrams, tables, charts, and references 2. Adequately addressing the topic and detailed information 3. Quality of demonstrated problems/examples 4. On-time submission, presentation, and overall understanding of the assigned topic. Additional requirements The report should be single-spaced with a 1-inch margin. The paper should use Times Roman 13-point font. The charts, diagrams, and tables must be neatly prepared, and the
report should be checked for proper grammar and spelling. There is no restriction on the number of pages and number of slides. Students who do not email their project will receive a grade of zero for the assignment. There is no makeup for missed projects. The deadline for project submission remains consistent for all students, including those enrolled in the DEN and in-class sections. DEN students are not required to physically attend classes or participate online in presenting their final project. Instead, they can email their project recording, PowerPoint (PPT), and report directly to me. Furthermore, I will email the project topics to ISE 514 In class and DEN students. Copying (including from past students’ projects) is not permitted. Please refer to the information on plagiarism in the USC Statement on Ethics for more information.

Planning and Scheduling in Manufacturing and Services Second edition
Planning and Scheduling in Manufacturing and Services Second edition Michael L. Pinedo
Allrightsreserved.permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenotidentiÞedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.Printedonacid-freepaperThisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenSpringer Dordrecht Heidelberg London New York Springer is part of Springer Science+Business Media (www.springer.com) Michael L. PinedoStern School of BusinessNew York University44 West 4th StreetNew York, NY 10012USAmpinedo@stern.nyu.eduLibraryofCongressControlNumber:MathematicsSubjectClassiÞcation(2000):90B06, 90B30, 90B35, 90B50ISBN978-1-4419-0909-1e-ISBN978-1-4419-0910-7DOI10.1007/978-1-4419-0910-72009930422DepartmentofInformation, Operations, ©SpringerScience+BusinessMedia,LLC2005,2009and Management Sciences
ToPaula,Esti,Jaclyn,andDanielle,EddieandJeffrey,Franciniti,Morris,andIzzy
PrefacePrefacetotheFirstEditionThisbookisanoutgrowthofanearliertextthatwasreleasedin1999un-derthetitle“OperationsSchedulingwithApplicationsinManufacturingandServices”,coauthoredwithXiuliChaofromNorthCarolinaState.Thisnewversionhasbeencompletelyreorganizedandexpandedinseveraldirectionsincludingnewapplicationareasandsolutionmethods.Theapplicationareasaredividedintotwoparts:manufacturingapplica-tionsandservicesapplications.Thebookcoversfiveareasinmanufacturing,namely,projectscheduling,jobshopscheduling,schedulingofflexibleassem-blysystems,economiclotscheduling,andplanningandschedulinginsupplychains.Itcoversfourareasinservices,namely,reservationsandtimetabling,tournamentscheduling,planningandschedulingintransportation,andwork-forcescheduling.Ofcourse,thisselectiondoesnotrepresentalltheapplica-tionsofplanningandschedulinginmanufacturingandservices.Someareasthathavereceivedafairamountofattentionintheliterature,e.g.,schedulingofroboticcells,havenotbeenincluded.Schedulingproblemsintelecommu-nicationandcomputersciencehavenotbeencoveredeither.Itseemshardertowriteagoodapplications-orientedbookthanagoodtheory-orientedbook.Inthewritingofthisbookonequestioncameupregu-larly:whatshouldbeincludedandwhatnot?Somedifficultdecisionshadtobemadewithregardtosomeofthematerialcovered.Forexample,shouldthisbookdiscussJohnson’srule,whichminimizesthemakespaninatwomachineflowshop?Johnson’sruleisdescribedinvirtuallyeveryschedulingbookandeveninmanybooksonoperationsmanagement.Itismathematicallyelegant;butitisnotclearhowimportantitisinpractice.Wefinallyconcludedthatitdidnotdeservesomuchattentioninanapplications-orientedbooksuchasthisone.However,wedidincorporateitasanexerciseinthechapteronjobshopschedulingandaskthestudenttocompareitsperformancetothatofthewell-knownshiftingbottleneckheuristic(whichisoneofthebetterknownheuristicsusedinpractice).
viiiPrefaceThefundamentalsconcerningthemethodologiesthatareusedintheap-plicationchaptersarecoveredintheappendixes.Theycontainthebasicsofmathematicalprogramming,dynamicprogramming,heuristics,andconstraintprogramming.Itisnotnecessarytohaveadetailedknowledgeofcomputationalcom-plexityinordertogothroughthisbook.However,attimessomecomplexityterminologyisused.Thatis,aschedulingproblemmaybereferredtoaspoly-nomiallysolvable(i.e.,easy)orasNP-hard(i.e.,hard).However,wenevergointoanyNP-hardnessproofs.Becauseofthediversityandthecomplexityofthemodels,itturnedouttobedifficulttodevelopanotationthatcouldbekeptuniformthroughoutthebook.Aseriousattempthasbeenmadetomaintainsomeconsistencyofnotation.However,thathasnotalwaysbeenpossible(but,ofcourse,withineachchapterthenotationisconsistent).Anotherissuewehadtodealwithwasthelevelofthemathematicalnotationused.Wedecidedthatwedidhavetoadoptattimesthesetnotationandusethe∈symbol.Soj∈SimpliesthatjobjbelongstoasetofjobscalledSandS1∪S2denotestheunionofthetwosetsS1andS2.ThebookcomeswithaCD-ROMthatcontainsvarioussetsofpowerpointslides.Fivesetsofslidesweredevelopedbyinstructorswhohadadoptedtheearlierversionofthisbook,namelyErwinHansandJohannHurinkatTwenteUniversityofTechnologyintheNetherlands,SiggiOlafssonatIowaState,SanjaPetrovicinNottingham,SibelSalmanatCarnegie-Mellon(Sibeliscur-rentlyatKo¸cUniversityinTurkey),andCeesDuinandErikvanderSluisattheUniversityofAmsterdam.Variouscollectionsofslideswerealsomadeavailablebyseveralcompanies,includingAlcan,CarmenSystems,Cybertec,DashOptimization,Ilog,Multimodal,andSAP.BothIlogandDashOpti-mizationprovidedasubstantialamountofadditionalmaterialintheformofsoftware,minicases,andamovie.TheCD-ROMalsocontainsvariousplanningandschedulingsystemsthathavebeendevelopedinacademia.TheLEKINsystemhasbeenespeciallydesignedforthemachineschedulingandjobshopmodelsdiscussedinChapter5.OthersystemsontheCD-ROMincludeacrewschedulingsystem,anemployeeschedulingsystem,andatimetablingsystem.Thisnewversionhasbenefitedenormouslyfromnumerouscommentsmadebymanycolleagues.Firstofall,thistextowesalottoXiuliChaofromNorthCarolinaState;hiscommentshavealwaysbeenextremelyuseful.Manyothershavealsogonethroughthemanuscriptandprovidedconstructivecriticisms.ThelistincludesYing-JuChen(NYU),JacquesDesrosiers(GERAD,Mon-treal),ThomasDong(ILOG),AndreasDrexl(Kiel,Germany),JohnFowler(Arizona),GuillermoGallego(Columbia),NicholasHall(OhioState),JackKanet(Clemson),Chung-YeeLee(HKUST),JosephLeung(NJIT),Haib-ingLi(NJIT),IrvLustig(ILOG),KirkMoehle(MaerskLine),DetlefPabst(Arizona),DenisSaure(UniversidaddeChile),ErikvanderSluis(UniversityofAmsterdam),MariusSolomon(NortheasternUniversity),ChelliahSriskan-darajah(UTDallas),MichaelTrick(Carnegie-Mellon),RehaUzsoy(Purdue),
PrefaceixAlkisVazacopoulos(DashOptimization),NitinVerma(DashOptimization),andBenjaminYen(HongKongUniversity).ThetechnicalproductionofthisbookandCD-ROMwouldnothavebeenpossiblewithoutthehelpofBernaSifonteandAdamLewenberg.ThanksarealsoduetotheNationalScienceFoundation;withoutitssupportthisprojectwouldnothavebeencompleted.Awebsiteforthisbookwillbemaintainedathttp://www.stern.nyu.edu/~mpinedoThissitewillkeepanup-to-datelistoftheinstructorswhoareusingthebook(includingthosewhousedthe1999version).Inaddition,thesitewillcontainrelevantmaterialthatbecomesavailableafterthebookhasgonetopress.NewYorkMichaelPinedoFall2004PrefacetotheSecondEditionThissecondeditionhasundergoneonemajorchangeandnumerousminorchanges.Themajorchangeinvolvesanewchapteronplanningandschedulinginhealthcare.OperationsResearchinhealthcarehasbeenaveryhottopicoverthelasttwodecadesandmanyresearchershavebegunpayingattentiontoitsinceitissuchanimportantcomponentofthenationalGDP.Moreover,inhealthcarethereareplanningandschedulingproblemsgalore.Withtheadditionofthisnewchapteronhealthcare,thebookcoversnowexactlytenapplicationareas,fiveinthemanufacturingpartandfiveintheservicespart.(Onedoeshavetotakeintoaccountthattheborderlinesbetweenmanufacturingandservicesareoftenblurry.)Allotherchaptershaveundergoneanumberofminorchangesaswell.Thesechangesusuallyinvolvecrossreferencesthatpointouttherelationshipsbetweenthevariousmodelsinthedifferentchapters.TheCD-ROMcontainssomenewmaterialaswell.ThefirsteditionofthebookhadbeenusedoverthelastcoupleofyearsasabasisforcoursesonplanningandschedulingattheUniversityofTorontobyChrisBeck,attheUniversityofSouthernDenmarkbyMarcoChiarandiniandattheUniversityofBonnbyTimNieberg.Chris,Marco,andTimhavedevelopedsomeverynicesetsofslidesfortheircoursesandaremakingthemavailabletoothers.TheirslidesarenowontheCD-ROMaswell.Thisedition,again,hasbenefitedenormouslyfromnumerouscommentsmadebymanycolleagues.ThelistofcolleaguesincludesDirkBriskorn(Uni-versityofKiel),MarcoChiarandini(UniversityofSouthernDenmark),CeesDuin(UniversityofAmsterdam),HeinrichKuhn(CatholicUniversityofEich-staett),TimNieberg(BonnUniversity),DetlefPabst(AMD),ErikvanderSluis(UniversityofAmsterdam).
xPrefaceAgain,thetechnicalproductionofthisbookandCD-ROMwouldnothavebeenpossiblewithoutthehelpofAdamLewenbergfromStanfordUniversity.IamverygratefultoAchiDosanjh(Springer)forhercontinuoussupport.ThanksarealsoduetotheNationalScienceFoundation;withoutitssupportthisprojectwouldnothavebeencompleted.Awebsiteforthisbookwillcontinuetobemaintainedathttp://www.stern.nyu.edu/~mpinedoThissitecontainscopiesofeightbookreviewsthathavebeenwrittenafterthefirsteditioncameout.Thesitewillkeepanup-to-datelistoftheinstructorswhoareusingthebook(includingthosewhousedthe1999versionandthe2005version).Inthefuture,thesitewillalsocontainrelevantmaterialthatbecomesavailableafterthesecondeditionhasappearedinprint.NewYorkMichaelPinedoSpring2009
ContentsPreface………………………………………………..viiContentsofCD-ROM…………………………………..xviiPartIPreliminaries1Introduction………………………………………..31.1PlanningandScheduling:RoleandImpact……………..31.2PlanningandSchedulingFunctionsinanEnterprise………81.3OutlineoftheBook………………………………..112ManufacturingModels……………………………….192.1Introduction……………………………………..192.2Jobs,Machines,andFacilities………………………..212.3ProcessingCharacteristicsandConstraints……………..242.4PerformanceMeasuresandObjectives………………….282.5Discussion……………………………………….323ServiceModels……………………………………..373.1Introduction……………………………………..373.2ActivitiesandResourcesinServiceSettings……………..403.3OperationalCharacteristicsandConstraints…………….413.4PerformanceMeasuresandObjectives………………….443.5Discussion……………………………………….46
xiiContentsPartIIPlanningandSchedulinginManufacturing4ProjectPlanningandScheduling……………………..534.1Introduction……………………………………..534.2CriticalPathMethod(CPM)………………………..564.3ProgramEvaluationandReviewTechnique(PERT)………604.4Time/CostTrade-Offs:LinearCosts…………………..634.5Time/CostTrade-Offs:NonlinearCosts………………..704.6ProjectSchedulingwithWorkforceConstraints…………..714.7ROMAN:AProjectSchedulingSystemfortheNuclearPowerIndustry……………………………………744.8Discussion……………………………………….785MachineSchedulingandJobShopScheduling………….835.1Introduction……………………………………..835.2SingleMachineandParallelMachineModels…………….845.3JobShopsandMathematicalProgramming……………..865.4JobShopsandtheShiftingBottleneckHeuristic………….895.5JobShopsandConstraintProgramming………………..955.6LEKIN:AGenericJobShopSchedulingSystem………….1045.7Discussion……………………………………….1116SchedulingofFlexibleAssemblySystems………………1176.1Introduction……………………………………..1176.2SequencingofUnpacedAssemblySystems………………1186.3SequencingofPacedAssemblySystems………………..1246.4SchedulingofFlexibleFlowSystemswithBypass…………1296.5MixedModelAssemblySequencingatToyota……………1346.6Discussion……………………………………….1377EconomicLotScheduling…………………………….1437.1Introduction……………………………………..1437.2OneTypeofItemandtheEconomicLotSize……………1447.3DifferentTypesofItems-RotationSchedules……………1487.4DifferentTypesofItems-ArbitrarySchedules…………..1527.5MoreGeneralELSPModels…………………………1617.6MultiproductPlanningandSchedulingatOwens-CorningFiberglas………………………………………..1647.7Discussion……………………………………….1668PlanningandSchedulinginSupplyChains…………….1738.1Introduction……………………………………..1738.2SupplyChainSettingsandConfigurations………………1758.3FrameworksforPlanningandSchedulinginSupplyChains…180
Contentsxiii8.4AMediumTermPlanningModelforaSupplyChain……..1868.5AShortTermSchedulingModelforaSupplyChain………1928.6CarlsbergDenmark:AnExampleofaSystemImplementation1958.7Discussion……………………………………….199PartIIIPlanningandSchedulinginServices9IntervalScheduling,Reservations,andTimetabling……..2079.1Introduction……………………………………..2079.2ReservationswithoutSlack………………………….2099.3ReservationswithSlack…………………………….2129.4TimetablingwithWorkforceConstraints……………….2159.5TimetablingwithOperatororToolingConstraints………..2189.6AssigningClassestoRoomsatU.C.Berkeley……………2249.7Discussion……………………………………….22610SchedulingandTimetablinginSportsandEntertainment.23110.1Introduction……………………………………..23110.2SchedulingandTimetablinginSportTournaments……….23210.3TournamentSchedulingandConstraintProgramming……..23910.4TournamentSchedulingandLocalSearch……………….24210.5SchedulingNetworkTelevisionPrograms……………….24510.6SchedulingaCollegeBasketballConference……………..24710.7Discussion……………………………………….25011Planning,Scheduling,andTimetablinginTransportation..25511.1Introduction……………………………………..25511.2TankerScheduling…………………………………25611.3AircraftRoutingandScheduling……………………..26011.4TrainTimetabling…………………………………27411.5JeppesenSystems:DesignandImplementation…………..28111.6Discussion……………………………………….28512PlanningandSchedulinginHealthCare……………….29112.1Introduction……………………………………..29112.2SchedulingaSingleOperatingRoom…………………..29212.3MultipleOperatingRooms-ASetPackingFormulation……29712.4MultipleOperatingRooms-AStochasticApproach………30112.5PlanningandSchedulingRadiotherapyTreatments……….30412.6EmergencyRoomStaffing-AConstraintProgrammingApproach………………………………………..30812.7ASurgerySchedulingandBedOccupancyLevellingSystem..31012.8Discussion……………………………………….313
xivContents13WorkforceScheduling………………………………..31713.1Introduction……………………………………..31713.2Days-OffScheduling……………………………….31813.3ShiftScheduling…………………………………..32413.4TheCyclicStaffingProblem…………………………32713.5ApplicationsandExtensionsofCyclicStaffing…………..32913.6CrewScheduling………………………………….33113.7OperatorSchedulinginaCallCenter………………….33513.8Discussion……………………………………….339PartIVSystemsDevelopmentandImplementation14SystemsDesignandImplementation…………………..34714.1Introduction……………………………………..34714.2SystemsArchitecture………………………………34814.3Databases,ObjectBases,andKnowledge-Bases………….35014.4ModulesforGeneratingPlansandSchedules…………….35514.5UserInterfacesandInteractiveOptimization…………….35814.6GenericSystemsvs.Application-SpecificSystems…………36414.7ImplementationandMaintenanceIssues………………..36715AdvancedConceptsinSystemsDesign………………..37315.1Introduction……………………………………..37315.2RobustnessandReactiveDecisionMaking……………..37415.3MachineLearningMechanisms……………………….37915.4DesignofPlanningandSchedulingEnginesandAlgorithmLibraries………………………………………..38515.5ReconfigurableSystems…………………………….38815.6Web-BasedPlanningandSchedulingSystems…………..39015.7Discussion……………………………………….39316WhatLiesAhead?…………………………………..39916.1Introduction……………………………………..39916.2PlanningandSchedulinginManufacturing……………..40016.3PlanningandSchedulinginServices…………………..40116.4SolutionMethods………………………………….40316.5SystemsDevelopment………………………………40516.6Discussion……………………………………….406
ContentsxvAppendicesAMathematicalProgramming:FormulationsandApplications411A.1Introduction……………………………………..411A.2LinearProgrammingFormulations…………………….411A.3NonlinearProgrammingFormulations………………….414A.4IntegerProgrammingFormulations……………………416A.5SetPartitioning,SetCovering,andSetPacking………….418A.6DisjunctiveProgrammingFormulations………………..419BExactOptimizationMethods…………………………423B.1Introduction……………………………………..423B.2DynamicProgramming……………………………..424B.3OptimizationMethodsforIntegerPrograms…………….428B.4ExamplesofBranch-and-BoundApplications……………430CHeuristicMethods…………………………………..441C.1Introduction……………………………………..441C.2BasicDispatchingRules…………………………….442C.3CompositeDispatchingRules………………………..445C.4BeamSearch……………………………………..449C.5LocalSearch:SimulatedAnnealingandTabu-Search………452C.6LocalSearch:GeneticAlgorithms……………………..459C.7Discussion……………………………………….461DConstraintProgrammingMethods…………………….465D.1Introduction……………………………………..465D.2ConstraintSatisfaction……………………………..466D.3ConstraintProgramming……………………………467D.4OPL:AnExampleofaConstraintProgrammingLanguage…469D.5ConstraintProgrammingvs.MathematicalProgramming…..472ESelectedSchedulingSystems………………………….475E.1Introduction……………………………………..475E.2GenericSystems…………………………………..475E.3Application-SpecificSystems…………………………476E.4AcademicPrototypes………………………………477
xviContentsFTheLekinSystemUser’sGuide………………………479F.1Introduction……………………………………..479F.2LinkingExternalAlgorithms…………………………479References……………………………………………..487Notation……………………………………………….519SubjectIndex………………………………………….523NameIndex……………………………………………529
ContentsofCD-ROM0.CDOverview1.SlidesfromAcademia(a)TwenteUniversity(byErwinHansandJohannHurink)(b)IowaStateUniversity(bySiggiOlafsson)(c)UniversityofNottingham(bySanjaPetrovic)(d)Carnegie-MellonUniversity(bySibelSalman)(e)UniversityofAmsterdam(byErikvanderSluisandCeesDuin)(f)UniversityofToronto(byChristopherBeck)(g)UniversityofBonn(byTimNieberg)(h)UniversityofSouthernDenmark(byMarcoChiarandini)2.SlidesfromCorporations(a)Alcan(b)JeppesenSystems(CarmenSystems)(c)Cybertec(d)FairIsaac(DashOptimization)(e)OliverWyman-Multimodal(f)SAP3.SchedulingSystems(a)LEKINJobShopSchedulingSystem
xviiiContentsofCD-ROM4.OptimizationSoftware(a)FairIsaac(DashOptimization)Software(withSampleProgramsforExamples8.4.1,11.2.1,11.3.1)(b)ILOGOPLSoftware(withSampleProgramsforExamples8.4.1and11.2.1)5.ExamplesandExercises(a)TankerScheduling(ComputationaldetailsofExample11.2.1)(b)AircraftRoutingandScheduling(ComputationaldetailsofExample11.3.1)6.Mini-cases(a)ILOG(b)FairIsaac(DashOptimization)7.AdditionalReadings(WhitePapers)(a)JeppesenSystems(CarmenSystems)(b)OliverWyman-Multimodal8.Movies(a)Saiga-SchedulingattheParisairports(ILOG)(b)UnitedAirlines-PegasusSoftware
PartIPreliminaries1Introduction………………………………….32ManufacturingModels………………………….193ServiceModels………………………………..37
Chapter1Introduction1.1PlanningandScheduling:RoleandImpact…..31.2PlanningandSchedulingFunctionsinanEnterprise……………………………..81.3OutlineoftheBook……………………..111.1PlanningandScheduling:RoleandImpactPlanningandschedulingareformsofdecision-makingthatareusedonaregularbasisinmanymanufacturingandserviceindustries.Decision-makingprocessesplayanimportantroleinprocurementandproduction,intrans-portationanddistribution,andininformationprocessingandcommunication.Theplanningandschedulingfunctionsinacompanyrelyonmathematicaltechniquesandheuristicmethodsthatallocatelimitedresourcestotheactiv-itiestobedone.Thisallocationofresourceshastobedoneinsuchawaythatthecompanyoptimizesitsobjectivesandachievesitsgoals.Resourcesmaybemachinesinaworkshop,runwaysatanairport,crewsataconstruc-tionsite,orprocessingunitsinacomputingenvironment.Activitiesmaybeoperationsinaworkshop,take-offsandlandingsatanairport,stagesinaconstructionproject,orcomputerprogramsthathavetobeexecuted.Eachactivitymayhaveaprioritylevel,anearliestpossiblestartingtimeand/oraduedate.Objectivescantakemanydifferentforms,suchasminimizingthetimetocompleteallactivities,minimizingthenumberofactivitiesthatarecompletedafterthecommittedduedates,andsoon.Thefollowingtenexamplesillustratetheroleofplanningandschedul-ingintherealworld.Eachexampledescribesaparticulartypeofplanningandschedulingproblem.Thefirstexampleshowstheroleofplanningandschedulinginthemanagementofalargeconstructionorinstallationprojectthatconsistsofmanystages.© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_1,3
41IntroductionExample1.1.1(ASystemInstallationProject).Considertheprocure-ment,installation,andtestingofalargecomputersystem.Theprojectin-volvesanumberofdistincttasks,includingevaluationandselectionofhard-ware,softwaredevelopment,recruitmentandtrainingofpersonnel,systemtesting,systemdebugging,andsoon.Aprecedencerelationshipstructureex-istsamongthesetasks:somecanbedoneinparallel(concurrently),whereasotherscanonlystartwhencertainpredecessorshavebeencompleted.Thegoalistocompletetheentireprojectintheshortesttime.Planningandschedulingdonotonlyprovideacoherentprocesstomanagetheproject,butalsoprovideagoodestimateforitscompletiontime,revealwhichtasksarecriticalanddeterminetheactualdurationoftheentireproject.Thesecondexampleistakenfromajobshopmanufacturingenvironment,wheretheimportanceofplanningandschedulingisgrowingwiththeincreas-ingdiversificationanddifferentiationofproducts.Thenumberofdifferentitemsthathavetobeproducedislargeandsetupcostsaswellasshippingdateshavetobetakenintoaccount.Example1.1.2(ASemiconductorManufacturingFacility).Semicon-ductorsaremanufacturedinhighlyspecializedfacilities.Thisisthecasewithmemorychipsaswellaswithmicroprocessors.Theproductionprocessinthesefacilitiesusuallyconsistsoffourphases:waferfabrication,waferprobe,assemblyorpackaging,andfinaltesting.Waferfabricationistechnologicallythemostcomplexphase.Layersofmetalandwafermaterialarebuiltupinpatternsonwafersofsiliconorgalliumarsenidetoproducethecircuitry.Eachlayerrequiresanumberofop-erations,whichtypicallyinclude:(i)cleaning,(ii)oxidation,depositionandmetallization,(iii)lithography,(iv)etching,(v)ionimplantation,(vi)photore-siststripping,and(vii)inspectionandmeasurement.Becauseitconsistsofmanylayers,eachwaferundergoestheseoperationsseveraltimes.Thus,thereisasignificantamountofrecirculationintheprocess.Wafersmovethroughthefacilityinlotsof24.Somemachinesmayrequiresetupstopreparethemforincomingjobs.Thesetuptimeoftendependsontheconfigurationsofthelotjustcompletedandthelotabouttostart.Thenumberofordersinthesystemisofteninthehundredsandeachhasitsownreleasedateandcommittedshippingorduedate.Thescheduler’sobjectiveistomeetasmanyofthecommittedshippingdatesaspossible,whilemaximizingthroughput.Thelattergoalisachievedbymaximizingequipmentutilization,especiallyofthebottleneckmachines.Hence,minimizationofidletimesandsetuptimesisalsorequired.Inmanymanufacturingenvironments,automatedmaterialhandlingsys-temsdictatetheflowofproductsthroughthesystem.Flexibleassemblysys-temsfallintothiscategory.Thescheduler’sjobinthiskindofenvironmentistodevelopthebestschedulewhilesatisfyingcertaintimingandsequencingconditions.Theschedulerthushaslessfreedominconstructingaschedule.Thenextillustrationdescribesaclassicalexampleofthistypeofenvironment.
1.1PlanningandScheduling:RoleandImpact5Example1.1.3(AnAutomobileAssemblyLine).Anautomobileas-semblylinetypicallyproducesmanydifferentmodels,allbelongingtoasmallnumberofcarfamilies.Forexample,thedifferentmodelswithinafamilymayincludeatwo-doorcoupe,afour-doorsedan,andastationwagon.Therearealsoanumberofdifferentcolorsandoptionpackages.Somecarshaveauto-matictransmissions,whileothersaremanual;somecarshavesunroofswhileothercarshavesolidroofs.Inanassemblylinetherearetypicallyseveralbottlenecks,wherethethroughputofaparticularmachineorprocessdeterminestheoverallpro-ductionrate.Thepaintshopisoftensuchabottleneck;everytimethecolorchangesthepaintgunshavetobecleaned,whichisatimeconsumingprocess.Oneoftheobjectivesistomaximizethethroughputbysequencingthecarsinsuchawaythattheworkloadateachstationisbalancedovertime.Thepreviousexamplesillustratesomeofthedetailedandshorttermas-pectsofplanningandschedulingprocesses.However,planningandschedulingoftenhavetodealwithmediumandlongtermissuesaswell.Example1.1.4(ProductionPlanninginaPaperMill).Theinputtoapapermilliswoodfiberandpulp;theoutputisfinishedrollsofpaper.Attheheartofthepapermillareitspapermachines,whichareverylargeandrepresentasignificantcapitalinvestment(between50and100milliondollarseach).Eachmachineproducesvarioustypesofpapercharacterizedbytheirbasisweights,gradesandcolors.Masterproductionplansforthesemachinesaretypicallydrawnuponanannualbasis.Theprojectedschedulesarecyclicwithcycletimesoftwoweeksorlonger.Aparticulartypeofpapermaybeproducedeithereverycycle,everyothercycle,orevenlessoften,dependinguponthedemand.Everytimethemachineswitchesoverfromonegradeofpapertoanother,asetupcostisincurred.Duringthechangeoverthemachinekeepsonproducingpaper.However,sincethepaperproducedduringachangeoverdoesnotmeetanyofthesetstandards,itiseithersoldatasteepdiscountorconsideredwasteandfedbackintotheproductionsystem.Theproductionplantriestomaximizeproduction,whileminimizingin-ventorycosts.Maximizingproductionimpliesminimizingchangeovertimes.Thismeanslongerproductionruns,whichinturnresultinhigherinventorycosts.Theoverallproductionplanisatrade-offbetweensetupcostsandin-ventorycosts.Eachoneofthefacilitiesdescribedinthelastthreeexamplesmaybelongtoanetworkoffacilitiesinwhichrawmaterialor(semi)finishedgoodsmovefromonefacilitytoanother;inafacilitytheproductiseitherbeingstoredormorevalueisbeingadded.Inmanyindustriestheplanningandschedulingofthesupplychainsarecrucial.Example1.1.5(PlanningandSchedulinginaSupplyChain).Con-siderthepapermillofthepreviousexample.Amillistypicallyanintegral
61Introductionpartofacomplexnetworkofproductionfacilitiesthatincludestimberland(wheretreesaregrownusingadvancedforestmanagementtechnology),pa-permillswheretherollsofpaperareproduced,convertingfacilitieswheretherollsaretransformedintopaperproducts(e.g.,bags,cartons,orcutsizepaper),distributioncenters(whereinventoryiskept)andend-consumersorretailers.Severaldifferentmodesoftransportationareusedbetweenthevar-iousstagesofthesupplychain,e.g.,trucks,trains,andbarges.Eachmodehasitsowncharacteristics,suchascost,speed,reliability,andsoon.Clearly,ineachstageofthesupplychainmorevalueisaddedtotheproductandthefurtherdownthesupplychain,themoreproductdifferentiationexists.Coordinatingtheentirenetworkisadauntingprocess.Theoverallgoalistominimizethetotalcostsincludingproductioncosts,transportationcostsandinventoryholdingcosts.Inmanymanufacturingenvironmentscustomershavecloserelationshipswiththemanufacturer.Thefactoryestablishesitsproductionscheduleincol-laborationwithitscustomersandmayallowthemtoreservemachinesforspecificperiodsoftime.Conceptually,theschedulingproblemofthemanu-facturerissimilartotheschedulingproblemsincarrentalagenciesandhotels,wherecarsandroomscorrespondtomachinesandtheobjectiveistomaximizetheutilizationoftheseresources.Example1.1.6(AReservationSystem).Acarrentalagencymaintainsafleetofvarioustypesofcars.Itmayhavefullsize,midsize,compact,andsubcompactcars.Somecustomersmaybeflexiblewithregardtothetypeofcartheyarewillingtorent,whileothersmaybeveryspecific.Acustomertypicallycallsintomakeareservationforcertaindaysandtheagencyhastodecidewhetherornottoprovidehim/herwithacar.Attimesitmaybeadvantageoustodenyacustomerareservationthatisforaveryshortperiodifthereisachancetorentthecarouttoanothercustomerforalongerperiod.Theagency’sobjectiveistomaximizethenumberofdaysitscarsarerentedout.Schedulingandtimetablingalsoplayanimportantroleinsportsandenter-tainment.Sporttournamentshavetobescheduledverycarefully.Theschedulehastobesuchthatalltheparticipatingteamsaretreatedfairlyandthatthepreferencesofthefansaretakenintoaccount.Timetablingplaysanimpor-tantroleinentertainmentaswell.Forexample,televisionprogramshavetobescheduledinsuchawaythatratings(andthereforeprofits)aremaximized.Aftertheprogramshavebeenassignedtotheirslots,thecommercialshavetobescheduledaswell.Example1.1.7(SchedulingaSoccerTournament).Considerasoccerleaguetournament.Thegameshavetobescheduledoverafixednumberofrounds.Animportantconsiderationinthecreationofascheduleisthat,ideally,eachteamshouldhaveaschedulethatalternatesbetweengamesat
1.1PlanningandScheduling:RoleandImpact7homeandgamesaway.However,itoftencannotbeavoidedthatateamhastoplaytwoconsecutivegamesathomeortwoconsecutivegamesaway.Therearemanyotherconcernsaswell:forexample,ifacityhastwoteamsparticipatinginthesameleague,thenitisdesirabletohaveineachroundoneteamathomeandtheotherteamaway.Iftwoteamsinaleagueareverystrong,thenitwouldbeniceifnoneoftheotherteamswouldhavetofacethesetwoteamsinconsecutiverounds.Planningandschedulingplayaveryimportantroleinthetransportationindustries.Therearevariousmodesoftransportationandthedifferentin-dustriesfocusondifferentwaysofmovingeithercargoorpassengers.Theobjectivesincludeminimizingtotalcostaswellasmaximizingconvenienceor,equivalently,minimizingpenaltycosts.Example1.1.8(RoutingandSchedulingofAirplanes).Themarketingdepartmentofanairlineusuallyhasaconsiderableamountofinformationwithregardtocustomerdemandforanygivenflight(aflightischaracterizedbyitsoriginanddestinationandbyitsscheduleddeparturetime).Basedonthedemandinformation,theairlinecanestimatetheprofitofassigningaparticulartypeofaircrafttoaflightlegunderconsideration.Theairlineschedulingproblembasicallyfocusesonhowtocombinethedifferentflightlegsintoso-calledround-tripsthatcanbeassignedtoaspecificplane.Aroundtripmaybesubjecttomanyconstraints:theturn-aroundtimeatanairportmustbelongerthanagivenminimumtime;acrewcannotbeondutyforadurationthatislongerthanwhattheFederalAviationAdministration(FAA)allows,andsoon.Planningandschedulingplayaveryimportantroleinhealthcareingen-eralandinhospitalsinparticular.Hospitalsandclinicsusuallyfacemanydifferentschedulingproblemsthatinvolvesurgeons,nurses,aswellasexpen-siveequipment.Theobjectiveisoftentheoptimizationoftheutilizationofscarceresources,suchas,forexample,operatingroomsorradiotherapyequip-ment.Example1.1.9(PlanningandSchedulingRadiotherapyTreatments).Inoncologydepartments,theplanningandschedulingofradiotherapysessionsplayanimportantroleinensuringthedeliveryoftherighttreatmenttopa-tients.Apatienttypicallyhastovisitatreatmentcenterseveraltimesaweekforagivennumberofconsecutiveweeksdependentuponhisorhertreatmentplan.Theschedulingofthetreatmentplansdealswiththeassignmentofpa-tientstotimeslotssubjecttomanyconstraints.Thegoalofsuchoutpatientschedulingistohaveanappointmentsystemthatoptimizesvariousperfor-mancemeasures.Animportantexampleofsuchaperformancemeasureisthemaximizationoftheutilizationoftheradiotherapyequipment.Inmanymanufacturingandserviceindustries,planningandschedulingoftenhavetodealwithresourcesotherthanmachines;themostimportantresource,besidesthemachinery,isusuallypersonnel.
81IntroductionExample1.1.10(SchedulingNursesinaHospital).Everyhospitalhasstaffingrequirementsthatchangefromdaytoday.Forinstance,thenumberofnursesrequiredonweekdaysisusuallymorethanthenumberrequiredonweekends,whilethestaffingrequiredduringthenightshiftmaybelessthanthatrequiredduringthedayshift.Stateandfederalregulationsandunionrulesmayprovideadditionalschedulingconstraints.Consequently,therearedifferenttypesofshiftpatterns,allwithdifferentcosts.Thegoalistodevelopshiftassignmentssothatalldailyrequirementsaremetandtheconstraintsaresatisfiedatminimalcost.Fromtheexamplesaboveitisclearthatplanningandschedulingisim-portantinmanufacturingaswellasinservices.Certaintypesofschedulingproblemsaremorelikelytooccurinmanufacturingsettings(e.g.,assemblylinescheduling),whileothersaremorelikelytooccurinservicesettings(e.g.,reservationsystems).Andcertaintypesofschedulingproblemsoccurinbothmanufacturingandservices;forexample,projectschedulingisimportantintheshipbuildingindustryaswellasinmanagementconsulting.Inmanyenvironmentsitmaynotbeimmediatelyclearwhatimpactplan-ningandschedulinghaveonanygivenobjective.Inpractice,thechoiceofscheduletypicallyhasameasurableimpactonsystemperformance.Indeed,animprovementinaschedulecanusuallycutdirectandindirectcostssignif-icantly,especiallyinacomplexproductionsetting.Unfortunately,planningandschedulingproceduresmaybedifficulttoim-plement.Theunderlyingmathematicaldifficultiesaresimilartothoseencoun-teredinotherbranchesofcombinatorialoptimization,whiletheimplementa-tiondifficultiesareoftencausedbyinaccuraciesinmodelrepresentationsorbyproblemsencounteredintheretrievalofdataandthemanagementofin-formation.Resolvingthesedifficultiestakesskillandexperience,butisoftenfinanciallyandoperationallywellworththeeffort.1.2PlanningandSchedulingFunctionsinanEnterprisePlanningandschedulingineitheramanufacturingoraserviceenvironmenthavetointeractwithmanyotherfunctions.Theseinteractionsaretypicallysystem-dependentandmaydiffersubstantiallyfromonesettingtoanother;theyoftentakeplacewithinacomputernetwork.Thereare,ofcourse,alsomanysituationswheretheexchangeofinformationbetweenplanningandschedulingandotherdecisionmakingfunctionsoccursinmeetingsorthroughmemos.PlanningandSchedulinginManufacturing.Wefirstdescribeagenericmanufacturingenvironmentandtheroleofitsplanningandschedul-ingfunction.Ordersthatarereleasedinamanufacturingsettinghavetobetranslatedintojobswithassociatedduedates.Thesejobsoftenhavetobeprocessedonthemachinesinaworkcenterinagivenorderorsequence.The
1.2PlanningandSchedulingFunctionsinanEnterprise9processingofjobsmaysometimesbedelayedifcertainmachinesarebusy.Preemptionsmayoccurwhenhighpriorityjobsarereleasedwhichhavetobeprocessedatonce.Unexpectedeventsontheshopfloor,suchasmachinebreakdownsorlonger-than-expectedprocessingtimes,alsohavetobetakenintoaccount,sincetheymayhaveamajorimpactontheschedules.Develop-ing,insuchanenvironment,adetailedscheduleofthetaskstobeperformedhelpsmaintainefficiencyandcontrolofoperations.Theshopfloorisnottheonlypartoftheorganizationthatimpactstheschedulingprocess.Theschedulingprocessalsointeractswiththeproductionplanningprocess,whichhandlesmedium-tolong-termplanningfortheentireorganization.Thisprocessintendstooptimizethefirm’soverallproductmixandlong-termresourceallocationbasedoninventorylevels,demandforecasts,andresourcerequirements.Decisionsmadeatthishigherplanninglevelmayimpactthemoredetailedschedulingprocessdirectly.Figure1.1depictsadiagramoftheinformationflowinamanufacturingsystem.Inmanufacturing,planningandschedulinghastointeractwithotherde-cisionmakingfunctionsintheplant.OnepopularsystemthatiswidelyusedistheMaterialRequirementsPlanning(MRP)system.Afteraschedulehasbeensetupitisnecessarythatalltherawmaterialsandresourcesareavail-ableatspecifiedtimes.ThereadydatesofthejobshavetobedeterminedbytheproductionplanningandschedulingsysteminconjunctionwiththeMRPsystem.MRPsystemsarenormallyfairlyelaborate.EachjobhasaBillOfMa-terials(BOM)itemizingthepartsrequiredforproduction.TheMRPsystemkeepstrackoftheinventoryofeachpart.Furthermore,itdeterminesthetim-ingofthepurchasesofeachoneofthematerials.Indoingso,itusestechniquessuchaslotsizingandlotschedulingthataresimilartothoseusedinplanningandschedulingsystems.TherearemanycommercialMRPsoftwarepackagesavailable.Asaresult,manymanufacturingfacilitiesrelyonMRPsystems.Inthecaseswherethefacilitydoesnothaveaplanningorschedulingsystem,theMRPsystemmaybeusedforproductionplanningpurposes.However,inacomplexsettingitisnoteasyforanMRPsystemtodothedetailedplanningandschedulingsatisfactorily.Modernfactoriesoftenemployelaboratemanufacturinginformationsys-temsinvolvingacomputernetworkandvariousdatabases.Localareanet-worksofpersonalcomputers,workstationsanddataentryterminalsarecon-nectedtoacentralserver,andmaybeusedeithertoretrievedatafromthevariousdatabasesortoenternewdata.Planningandschedulingisusuallydoneononeofthesepersonalcomputersorworkstations.Terminalsatkeylocationsmayoftenbeconnectedtotheschedulingcomputerinordertogivedepartmentsaccesstocurrentschedulinginformation.Thesedepartments,inturn,mayprovidetheschedulingsystemwithrelevantinformation,suchaschangesinjobstatus,machinestatus,orinventorylevels.CompaniesnowadaysoftenrelyonelaborateEnterpriseResourcePlanning(ERP)systems,thatcontrolandcoordinatetheinformationinallitsdivisions
101IntroductionProduction planning,master schedulingShop-floormanagementDispatchingMaterial requirements,planningcapacity planningSchedulingandreschedulingShop-floorCapacitystatusSchedulingconstraintsScheduleperformanceShopstatusDatacollectionJob loadingDetailedschedulingMaterialrequirementsOrders,demand forecastsQuantities,due datesScheduleShop orders,release datesFig.1.1.Informationflowdiagraminamanufacturingsystem
1.3OutlineoftheBook11andsometimesalsoatitssuppliersandcustomers.DecisionsupportsystemsofvariousdifferenttypesmaybelinkedtosuchanERPsystem,enablingthecompanytodolongrangeplanning,mediumtermplanningaswellasshorttermscheduling.PlanningandSchedulinginServices.Describingagenericserviceor-ganizationanditsplanningandschedulingsystemsisnotasstraightforwardasdescribingagenericmanufacturingsystem.Theplanningandschedulingfunctionsinaserviceorganizationoftenfacemanydifferentproblems.Theymayhavetodealwiththereservationofresources(e.g.,trucks,timeslots,meetingroomsorotherresources),theallocation,assignment,andschedul-ingofequipment(e.g.,specializedequipment,planes)ortheallocationandschedulingoftheworkforce(e.g.,theassignmentofshiftsinacallcenter).Thealgorithmstendtobecompletelydifferentfromthoseusedinmanufac-turingsettings.Planningandschedulinginaserviceenvironmentalsohavetointeractwithotherdecisionmakingfunctions,usuallywithinelaboratein-formationsystems,muchinthesamewayastheschedulingfunctioninamanufacturingsetting.Theseinformationsystemstypicallyrelyonextensivedatabasesthatcontainalltherelevantinformationregardingtheavailabilityofresourcesaswellasthecharacteristicsofcurrentandpotentialcustomers.Aplanningandschedulingsystemmayinteractwithaforecastingmodule;itmayalsointeractwithayieldmanagementmodule(whichisatypeofmodulenotverycommoninmanufacturingsettings).Ontheotherhand,inaserviceenvironmentthereisusuallynoMRPsystem.Figure1.2depictstheinformationflowinaserviceorganizationsuchasacarrentalagency.1.3OutlineoftheBookThisbookfocusesonplanningandschedulingapplications.Althoughthou-sandsofplanningandschedulingmodelsandproblemshavebeenstudiedintheliterature,onlyalimitednumberareconsideredinthisbook.Theselec-tionisbasedontheinsightthemodelsprovide,themethodologiesneededfortheiranalysesandtheirimportancewithregardtoreal-worldapplica-tions.Thisbookconsistsoffourparts.PartIdescribesthegeneralcharacteris-ticsofschedulingmodelsinmanufacturingandinservices.PartIIconsidersvariousclassesofplanningandschedulingmodelsinmanufacturing,andPartIIIdiscussesseveralclassesofplanningandschedulingmodelsinservices.PartIVdealswithsystemdesign,development,andimplementationissues.TheremainderofPartIconsistsofChapters2and3.Chapter2discussesthebasiccharacteristicsofthemanufacturingmodelsthatareconsideredinPartIIofthisbookandChapter3describesthecharacteristicsoftheservicemodelsthatareconsideredinPartIII.Thesecharacteristicsincludemachineenvironmentsandservicesettings,processingrestrictionsandconstraints,aswellasperformancemeasuresandobjectivefunctions.
121IntroductionDatabaseForecastingSchedulingYieldmanagementCustomerForecastsStatus/historyPrices/rulesPlace order/make reservationsAccept/reject(conditions)DataFig.1.2.InformationflowdiagraminaservicesystemPartIIfocusesonplanningandschedulinginmanufacturingandconsistsofChapters4to8.Eachoneofthesechaptersfocusesonadifferentclassofplanningandschedulingmodelswithapplicationsinmanufacturing;eachchaptercorrespondstooneoftheexamplesdiscussedinSection1.1.Atfirst,itmayappearthatthevariouschaptersinthispartaresomewhatunrelatedtoeachotherandareselectedarbitrarily.However,thereisarationalebehindtheselectionaswellasthesequenceofthetopics;thechaptersareactuallycloselyrelatedtoeachother.Chapter4focusesonprojectscheduling.Aprojectschedulingproblemusuallyconcernsasingleprojectthatconsistsofanumberofseparatejobsthatarerelatedtooneanotherthroughprecedenceconstraints.Sinceonlyasingleprojectisconsidered,thebasicformatofthistypeofschedulingproblemisinherentlyeasy,andthereforethelogicalonetostartoutwith.Moreover,animmediategeneralizationofthisproblem,i.e.,theprojectschedulingproblemwithworkforceconstraints,hasfromamathematicalpointofviewseveral
1.3OutlineoftheBook13importantspecialcases.Forexample,thejobshopschedulingproblemsdis-cussedinChapter5andthetimetablingproblemsconsideredinChapter9aresuchspecialcases.Chapter5coverstheclassicalsinglemachine,parallelmachine,andjobshopschedulingmodels.Inasinglemachineaswellasinaparallelmachineenvironment,ajobconsistsofasingleoperation;inaparallelmachineenvi-ronmentthisoperationmaybedoneonanyoneofthemachinesavailable.Inajobshop,eachjobhastoundergomultipleoperationsonthevariousmachinesandeachjobhasitsownsetofprocessingtimesandroutingcharacteristics.Severalobjectivesareofinterest;themostimportantoneisthemakespan,whichisthetimerequiredtofinishalljobs.Chapter6focusesonflexibleassemblysystems.Thesesystemshavesomesimilaritieswithjobshops;however,therearealsosomedifferences.Inaflexibleassemblysystemthereareseveraldifferentjobtypes;but,incontrasttoajobshop,acertainnumberhastobeproducedofeachtype.Theroutingconstraintsinflexibleassemblysystemsarealsosomewhatdifferentfromthoseinjobshops.Becauseofthepresenceofamaterialhandlingorconveyorsystem,thestartingtimeofoneoperationmaybeaveryspecificfunctionofthecompletiontimeofanotheroperation.(Injobshopsthesedependenciesareconsiderablyweaker.)TheproblemsconsideredinChapter7,lotsizingandscheduling,aresome-whatsimilartothoseinChapter6.Thereareagainvariousdifferentjobtypes,andofeachtypethereareanumberofidenticaljobs.However,thevarietyofjobtypesinthischapterisusuallylessthanthevarietyofjobtypesinaflexibleassemblysystem.Thenumbertobeproducedofanyparticularjobtypetendstobelargerthaninaflexibleassemblysystem.Thisnumberiscalledthelotsizeanditsdeterminationisanintegralpartoftheschedulingproblem.So,goingfromChapter5toChapter7,thevarietyintheorderportfoliodecreases,whilethebatchorlotsizesincrease.Chapter8focusesonplanningandschedulinginsupplychains.Thischap-terassumesthatthemanufacturingenvironmentconsistsofanetworkofrawmaterialandpartsproviders,productionfacilities,distributioncenters,cus-tomers,andsoon.Theproductflowsfromonestagetothenextandateachstagemorevalueisaddedtotheproduct.Eachfacilitycanbeoptimizedlo-callyusingtheproceduresthataredescribedinChapters5,6and7.However,performingaglobaloptimizationthatencompassestheentirenetworkrequiresaspecialframework.Suchaframeworkhastotakenowalsotransportationissuesintoaccount;transportationcostsappearintheobjectivefunctionandthequantitiestobetransportedbetweenfacilitiesmaybesubjecttorestric-tionsandconstraints.PartIIIfocusesonplanningandschedulinginservicesandconsistsofChapters9to13.Eachchapterdescribesaclassofplanningandschedulingmodelsinagivenservicesetting,andeachchaptercorrespondstooneoftheexamplesdiscussedinSection1.1.ThetopicscoveredinPartIIIare,clearly,differentfromthoseinPartII.However,Chapters9to13alsodealwithissues
141Introductionandfeaturesthatmayalsobeimportantinmanufacturingsettings.However,addingthesefeaturestothemodelsdescribedinChapters4to8wouldleadtoproblemsthatareextremelyhardtoanalyze.ThatiswhythesefeaturesareconsideredseparatelyinrelativelysimplesettingsinPartIII.Chapter9considersreservationsystemsandtimetablingmodels.Theseclassesofmodelsarebasicallyequivalenttoparallelmachinemodels.Inreser-vationmodels,jobs(i.e.,reservations)tendtohavereleasedatesandduedatesthataretight;thedecision-makerhastodecidewhichjobstoprocessandwhichjobsnottoprocess.Reservationmodelsareimportantinhospital-ityindustriessuchashotelsandcarrentalagencies.Intimetablingmodelsthejobsaresubjecttoconstraintswithregardtotheavailabilityofoperatorsortools.Timetablingmodelsarealsoimportantintheschedulingofmeetings,classes,andexams.Chapter10describesschedulingandtimetablinginsportsandentertain-ment.Theschedulingoftournaments(e.g.,basketball,baseball,soccer,andsoon)tendstobeverydifficultbecauseofthemanypreferencesandconstraintsconcerningtheschedules.Forexample,itisdesirablethatthesequenceofgamesassignedtoanygiventeamalternatesbetweengamesathomeandgamesaway.Thischapterdiscussesanoptimizationapproach,aconstraintprogrammingapproach,aswellasalocalsearchapproachfortournamentschedulingproblems.Italsodescribeshowtoscheduleprogramsinbroadcasttelevisionsoastomaximizetheratings.Chapter11discussesplanning,schedulingandtimetablingintransporta-tionsettings.Thetransportationsettingsincludetheschedulingofoiltankers,theroutingandschedulingofairplanes,andthetimetablingoftrains.Tankershavetomovecargoesfromonepointtoanotherwithingiventimewindowsandairlineshavetocoverflightlegsbetweencertaincitiesalsowithingiventimewindows.Oneimportantobjectiveistoassignthedifferenttripsorflightlegstothegiventankersorairplanesinsuchawaythatthetotalcostisminimized.Thereareimportantsimilaritiesaswellasdifferencesbetweentheschedulingofoiltankers,theroutingandschedulingofaircraft,andthetimetablingoftrains.Chapter12considersplanningandschedulinginhealthcare.Therearemanydifferenttypesofplanningandschedulingproblemsinthehealthcarein-dustry.Oneimportantclassofschedulingproblemsinvolvestheschedulingofplanned(elective)surgeriesinoperatingtheatres.Anotherclassofschedulingproblemsconcernsappointmentsystemsintreatmentcenters.Theproblemsdiscussedinthischapterarecloselyrelatedtotheparallelmachineschedul-ingproblemsconsideredinChapter5aswellastothetimetablingproblemsdiscussedinChapter9.Chapter13focusesonworkforcescheduling.Therearevariousdifferentclassesofworkforceschedulingmodels.Oneclassofworkforceschedulingmodelsincludestheshiftschedulingofnursesinhospitalsoroperatorsincallcenters.Adifferentclassofworkforceschedulingmodelsincludescrewschedulinginairlinesortruckingcompanies.
1.3OutlineoftheBook15Appendix D Constraint Programming MethodsAppendix C Heuristic MethodsAppendix B Exact Optimization MethodsAppendix A Mathematical Programming FormulationsAppendix E Selected Scheduling SystemsAppendixF LekinSystem Users Guide4. Project Planning and Scheduling5. Machine Scheduling and Job Shop Scheduling6. Scheduling of Flexible Assembly Systems7.Economic Lot Scheduling8. Planning and Scheduling in Supply Chains9. Interval Scheduling, Reservations and Timetabling10. Planning and Scheduling in Sports and Entertainment11.Planning, Scheduling and Timetabling in Transportation12.Planning and Scheduling in Health Care13.Workforce Scheduling1. Introduction3. Service Models2. Manufacturing Models16.What Lies Ahead?15. Advanced Concepts in Systems Design14.Systems Design and ImplementationFig.1.3.OverviewoftheBook
161IntroductionPartIVconcernssystemsdevelopmentandimplementationissuesandconsistsofthethreeremainingchapters.Chapter14dealswiththedesign,development,andimplementationofplanningandschedulingsystems.Itcov-ersbasicissueswithregardtosystemarchitecturesanddescribesthevari-oustypesofdatabases,planningandschedulingengines,anduserinterfaces.Thedatabasesmaybeconventionalrelationaldatabasesormoresophisti-catedknowledge-bases;theuserinterfacesmayincludeGanttcharts,capacitybuckets,orthroughputdiagrams.Chapter15describesmoreadvancedconceptsinsystemsdesign.Thetop-icsdiscussedincluderobustnessissues,learningmechanisms,systemsreconfig-urability,aswellasInternetrelatedfeatures.Thesetopicshavebeeninspiredbythedesign,developmentandimplementationofplanningandschedulingsystemsduringthelastdecadeofthetwentiethcentury;inrecentyearstheseconceptshavereceivedafairamountofattentionintheacademicliterature.Chapter16,thelastchapterofthisbook,discussesfuturedirectionsintheresearchanddevelopmentofplanningandschedulingapplications.Issuesinmanufacturingapplicationsarediscussed(whichoftenconcerntopicsinsup-plychainmanagement),issuesinserviceapplicationsarecovered,aswellasissuesinthedesignanddevelopmentofcomplexsystemsthatconsistofmanydifferentmodules.Importantissuesconcerntheconnectivityandinterfacedesignbetweenthedifferentmodulesofanintegratedsystem.Therearesixappendixes.AppendixAcoversmathematicalprogrammingformulations.AppendixBfocusesonexactoptimizationmethods,includ-ingdynamicprogrammingandbranch-and-bound.AppendixCdescribesthemostpopularheuristictechniques,e.g.,dispatchingrules,beamsearch,aswellaslocalsearch.AppendixDcontainsaprimeroncontraintprogramming.Ap-pendixEpresentsanoverviewofselectedschedulingsystemsandAppendixFprovidesauser’sguidetotheLEKINjobshopschedulingsystem.Figure1.3showstheprecedencerelationshipsbetweenthevariouschap-ters.Someoftheseprecedencerelationshipsarestrongerthanothers.Thestrongerrelationshipsaredepictedbysolidarrowsandtheweakeronesbydottedarrows.Thisbookisintendedforaseniorormasterslevelcourseonplanningandschedulingineitheranengineeringorabusinessschool.Theselectionofchapterstobecovereddepends,ofcourse,ontheinstructor.ItappearsthatinmostcasesChapters1,2and3havetobecovered.However,aninstructormaynotwanttogothroughallofthechaptersinPartsIIorIII;heorshemayselect,forexample,mostofChapters4and5andasmatteringofChapters6to12.AninstructormaydecidenottocoverPartIV,butstillassignthosechaptersasbackgroundreading.PrerequisiteknowledgeforthisbookisanelementarycourseinOperationsResearchonthelevelofHillierandLieberman’sIntroductiontoOperationsResearch.
CommentsandReferences17CommentsandReferencesOverthelastthreedecadesmanyschedulingbookshaveappeared,rangingfromtheelementarytotheadvanced.MostofthesefocusonjustoneortwoofthetenmodelcategoriescoveredinPartsIIandIIIofthisbook.Severalbooksarecompletelydedicatedtoprojectscheduling;see,forexample,ModerandPhilips(1970),Kerzner(1994),Kolisch(1995),Neumann,Schwindt,andZimmermann(2001),andDemeulemeesterandHerroelen(2002).Therearealsovariousbooksthathaveacoupleofchaptersdedicatedtothistopic(see,forexample,MortonandPentico(1993)).AsurveypaperbyBrucker,Drexl,M¨ohring,Neumann,andPesch(1999)presentsadetailedoverviewofthemodelsaswellasthetechniquesusedinprojectscheduling.Manybooksemphasizejobshopscheduling(oftenalsoreferredtoasmachinescheduling).OneofthebetterknowntextbooksonthistopicistheonebyCon-way,MaxwellandMiller(1967)(which,althoughslightlyoutofdate,isstillveryinteresting).AmorerecenttextbyBaker(1974)givesanexcellentoverviewofthemanyaspectsofmachinescheduling.AnintroductorytextbookbyFrench(1982)coversmostofthetechniquesthatareusedinjobshopscheduling.Themoread-vancedbookbyBlazewicz,Cellary,SlowinskiandWeglarz(1986)focusesmainlyonjobshopschedulingwithresourceconstraintsandmultipleobjectives.ThebookbyBlazewicz,Ecker,SchmidtandWeglarz(1993)isalsosomewhatadvancedanddealsprimarilywithcomputationalandcomplexityaspectsofmachineschedulingmod-elsandtheirapplicationstomanufacturing.ThemoreappliedtextbyMortonandPentico(1993)presentsadetailedanalysisofalargenumberofschedulingheuris-ticsthatareusefulforpractitioners.ThesurveypaperbyLawler,Lenstra,RinnooyKanandShmoys(1993)givesadetailedoverviewofmachinescheduling.Recently,anumberofbookshaveappearedthatfocusprimarilyonmachineschedulingandjobshopscheduling,seeDauz`ere-P´er`esandLasserre(1994),Baker(1995),Parker(1995),OvacikandUzsoy(1997),Sule(1996),Bagchi(1999),Brucker(2004),andPinedo(1995,2002,2008).ThebookseditedbyChr´etienne,Coffman,LenstraandLiu(1995)andLeeandLei(1997)containawidevarietyofpapersonmachinescheduling.Baptiste,LePapeandNuijten(2001)covertheapplicationofconstraintprogrammingtechniquestojobshopscheduling.ThevolumeeditedbyNareyek(2001)containspapersonlocalsearchappliedtojobshopscheduling.Severalbooksfocusonflexibleassemblysystems,flexiblemanufacturingsys-tems,orintelligentsystemsingeneral;see,forexample,Kusiak(1990,1992).Thesebooksusuallyincludeoneormorechaptersthatdealwithschedulingaspectsofthesesystems.Scholl(1998)focusesinhisbookonthebalancingandsequencingofassemblylines.Lotsizingandschedulingiscloselyrelatedtoinventorytheory.Haase(1994)givesanexcellentoverviewofthisfield.Br¨uggemann(1995)andKimms(1997)providemoreadvancedtreatments.DrexlandKimms(1997)provideanexhaustivesurveywithextensions.Lotschedulinghasalsobeencoveredinanumberofsurveypapersonproductionscheduling;see,forexample,thereviewsbyGraves(1981)andRodammerandWhite(1988).Planningandschedulinginsupplychainshavereceivedafairamountofattentionoverthelastdecade.Anumberofbookscoversupplychainsingeneral.Someoftheseemphasizetheplanningandschedulingaspects;see,forexample,Shapiro
181Introduction(2001)andMiller(2002).StadtlerandKilger(2002)editedabookwithvariouschaptersonplanningandschedulinginsupplychains.Intervalscheduling,reservationsystemsandtimetablingarefairlynewtopicsthatarecloselyrelatedtooneanother.IntervalschedulinghasbeenconsideredinacoupleofchaptersinDempster,LenstraandRinnooyKan(1982)andtimetablinghasbeendiscussedinthetextbookbyParker(1995).Overthelastdecade,aseriesofproceedingsofconferencesontimetablingappeared;seeBurkeandRoss(1996),BurkeandCarter(1998),BurkeandErben(2001),BurkeandDeCausmaecker(2003),BurkeandTrick(2004),andBurkeandRudova(2006).Afairamountofresearchhasbeendoneonplanningandschedulinginsportsandentertainment.Asfarasthisauthorisaware,nobookhasappearedthatiscom-pletelydedicatedtothisarea.However,thebookeditedbyButenko,Gil-LafuenteandPardalos(2004)hasseveralchaptersthatfocusonplanningandschedulinginsports.Noframeworkhasyetbeenestablishedforplanningandschedulingmodelsintransportation.Aseriesofconferencesoncomputer-aidedschedulingofpublictrans-porthasresultedinanumberofinterestingproceedings,seeWrenandDaduna(1988),DesrochersandRousseau(1992),Daduna,Branco,andPintoPaixao(1995),Wilson(1999),andVossandDaduna(2001).AHandbookeditedbyBarnhartandLaporte(2006)containschaptersthatcoverplanningandschedulinginmaritimeapplications,aviationandrailtransport.AvolumeeditedbyYu(1998)considersoperationsresearchapplicationsintheairlineindustry;thisvolumecontainsseveralpapersonplanningandschedulinginthatindustry.Aconsiderableamountofworkhasbeendoneonplanningandschedulinginhealthcare.Brandeau,SainfortandPierskalla(2004)editedahandbookinOpera-tionsResearchandhealthcare;severalchaptersinthishandbookdealwithplanningandschedulingapplications.Personnelschedulinghasnotreceivedthesameamountofattentionasprojectschedulingorjobshopscheduling.TienandKamiyama(1982)presentanoverviewonmanpowerschedulingalgorithms.ThetextbyNandaandBrowne(1992)iscom-pletelydedicatedtopersonnelscheduling.TheHandbookofIndustrialEngineeringhasachapteronpersonnelschedulingbyBurgessandBusby(1992).Parker(1995)dedicatesasectiontostaffingproblems.Burke,DeCausmaecker,VandenBergheandVanLandeghem(2004)presentanoverviewofthestateoftheartinnurserostering.Severalofthebooksonplanningandschedulingintransportationhavechaptersoncrewscheduling.Recently,aHandbookofSchedulingappeared,editedbyLeung(2004).Thishandbookcontainsnumerouschapterscoveringawidespectrumthatincludesjobshopscheduling,timetabling,tournamentscheduling,andworkforcescheduling.Anenormousamountofworkhasbeendoneonthedesign,developmentandimplementationofschedulingsystems.Mostofthisresearchanddevelopmenthasbeendocumentedinsurveypapers.Atabakhsh(1991)presentsasurveyofconstraintbasedschedulingsystemsusingartificialintelligencetechniquesandNoronhaandSarma(1991)giveanoverviewofknowledge-basedapproachesforschedulingprob-lems.Smith(1992)focusesinhissurveyonthedevelopmentandimplementationofschedulingsystems.Twocollectionsofpapers,editedbyZwebenandFox(1994)andbyBrownandScherer(1995),describeanumberofschedulingsystemsandtheiractualimplementation.
Chapter2ManufacturingModels2.1Introduction……………………………192.2Jobs,Machines,andFacilities……………..212.3ProcessingCharacteristicsandConstraints…..242.4PerformanceMeasuresandObjectives………282.5Discussion……………………………..322.1IntroductionManufacturingsystemscanbecharacterizedbyavarietyoffactors:thenum-berofresourcesormachines,theircharacteristicsandconfiguration,thelevelofautomation,thetypeofmaterialhandlingsystem,andsoon.Thedif-ferencesinallthesecharacteristicsgiverisetoalargenumberofdifferentplanningandschedulingmodels.Inamanufacturingmodel,aresourceisusu-allyreferredtoasa“machine”;ataskthathastobedoneonamachineistypicallyreferredtoasa“job”.Inaproductionprocess,ajobmaybeasingleoperationoracollectionofoperationsthathavetobedoneonvariousdifferentmachines.BeforedescribingthemaincharacteristicsoftheplanningandschedulingproblemsconsideredinPartIIofthisbook,wegiveabriefoverviewoffiveclassesofmanufacturingmodels.Thefirstclassofmodelsaretheprojectplanningandschedulingmodels.Projectplanningandschedulingisimportantwheneveralargeproject,thatconsistsofmanystages,hastobecarriedout.Aproject,suchasthecon-structionofanaircraftcarrieroraskyscraper,typicallyconsistsofanumberofactivitiesorjobsthatmaybesubjecttoprecedenceconstraints.Ajobthatissubjecttoprecedenceconstraintscannotbestarteduntilcertainotherjobshavebeencompleted.Inprojectscheduling,itisoftenassumedthatthereareanunlimitednumberofmachinesorresources,sothatajobcanstartassoon© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_2,19
202ManufacturingModelsasallitspredecessorshavebeencompleted.Theobjectiveistominimizethecompletiontimeofthelastjob,commonlyreferredtoasthemakespan.Itisalsoimportanttofindthesetofjobsthatdeterminesthemakespan,asthesejobsarecriticalandcannotbedelayedwithoutdelayingthecompletionoftheentireproject.Projectschedulingmodelsarealsoimportantintheplanningandschedulingofservices.Consider,forexample,theplanningandschedulingofalargeconsultingproject.Thesecondclassofmodelsincludesinglemachine,parallelmachineandjobshopmodels.Inasinglemachineorparallelmachineenvironment,ajobconsistsofoneoperationthatcanbedoneonanyoneofthemachinesavail-able.Inafull-fledgedjobshop,ajobtypicallyconsistsofanumberofopera-tionsthathavetobeperformedondifferentmachines.Eachjobhasitsownroutethatithastofollowthroughthesystem.Theoperationsofthejobsinajobshophavetobescheduledtominimizeoneormoreobjectives,suchasthemakespanorthenumberoflatejobs.Jobshopsareprevalentinindus-triesthatmakecustomizedindustrialhardware.However,theyalsoappearinserviceindustries(e.g.,hospitals).Aspecialcaseofajobshopisasettingwhereeachoneofthejobshastofollowthesameroutethroughthesystem(i.e.,eachjobhastobeprocessedfirstonmachine1,thenonmachine2,andsoon);suchasettingisusuallycalledaflowshop.Thethirdclassofmodelsfocusesonproductionsystemswithautomatedmaterialhandling.Inthesesettingsajobalsoconsistsofanumberofop-erations.Amaterialhandlingorconveyorsystemcontrolsthemovementofthejobsaswellasthetimingoftheirprocessingonthevariousmachines.Examplesofsuchenvironmentsareflexiblemanufacturingsystems,flexibleassemblysystems,andpacedassemblylines.Theobjectiveistypicallytomaximizethroughput.Suchsettingsareprevalentintheautomotiveindustryandintheconsumerelectronicsindustry.Thefourthclassofmodelsareknownaslotschedulingmodels.Thesemod-elsareusedformediumandlongtermproductionplanning.Incontrasttothefirstthreeclasses,theproductionanddemandprocessesarenowcontin-uous.Inthisclass,thereareavarietyofdifferentproducts.Whenamachineswitchesfromoneproducttoanother,achangeovercostisincurred.Thegoalisusuallytominimizetotalinventoryandchangeovercosts.Thesemodelsareimportantintheprocessindustries,suchasoilrefineriesandpapermills.Thefifthclassofmodelsconsistsofsupplychainplanningandschedulingmodels.Thesemodelstendtobehierarchicalandareoftenbasedonaninte-grationofthelotschedulingmodels(thefourthclassofmodels)andthejobshopschedulingmodels(thesecondclassofmodels).Theobjectivefunctionsinsupplychainplanningandschedulingtakeintoaccountinventoryhold-ingcostsatthevariousstagesinthechainaswellascostsoftransportationbetweenthestages.Therearerestrictionsandconstraintsontheproductionquantitiesaswellasonthequantitiesthathavetobetransportedfromonestagetoanother.
2.2Jobs,Machines,andFacilities21Themanufacturingmodelsdescribedabovecanbeclassifiedaseitherdis-creteorcontinuous.Theprojectschedulingmodels,jobshopmodels,andflexibleassemblysystemsarediscretemodels.Thelotschedulingmodelsarecontinuous.Themodelsforsupplychainplanningandschedulingcanbeei-thercontinuousordiscrete.Adiscretemodelcanusuallybeformulatedasanintegerprogramordisjunctiveprogram,whereasacontinuousmodelcanbeformulatedasalinearornonlinearprogram(seeAppendixA).Therearesimilaritiesaswellasdifferencesbetweenthemanufacturingmodelsinthesecondpartofthisbookandtheservicemodelsinthethirdpart.Oneclassofmodels,namelytheprojectschedulingmodels,isimportantforbothmanufacturingandservices.2.2Jobs,Machines,andFacilitiesThefollowingterminologyandnotationisusedthroughoutPartIIofthebook.Thenumberofjobsisdenotedbynandthenumberofmachinesbym.Thesubscriptsjandkrefertojobsjandk.Thesubscriptshandirefertomachineshandi.Thefollowingdatapertaintojobj.Processingtime(pij)Theprocessingtimepijrepresentsthetimejobjhastospendonmachinei.Thesubscriptiisomittediftheprocessingtimeofjobjdoesnotdependonthemachineorifitonlyneedsprocessingononemachine.Ifthereareanumberofidenticaljobsthatallneedaprocessingtimepjononemachine,thenwerefertothissetofjobsasitemsoftypej.Themachine’sproductionrateoftypejitemsisdenotedbyQj=1/pj(numberofitemsperunittime).Releasedate(rj).Thereleasedaterjofjobjisalsoknownasthereadydate.Itisthetimethejobarrivesatthesystem,i.e.,theearliesttimeatwhichjobjcanstartitsprocessing.Duedate(dj).Theduedatedjofjobjrepresentsthecommittedshippingorcompletiondate(thedatethejobispromisedtothecustomer).Completionofajobafteritsduedateisallowed,butapenaltyisthenincurred.Whentheduedateabsolutelymustbemet,itisreferredtoasadeadline.Weight(wj).Theweightwjofjobjisapriorityfactor,reflectingtheimportanceofjobjrelativetootherjobsinthesystem.Itmayrepresentthecostofkeepingjobjinthesystemforonetimeunit.Theweightcanbeaholdingorinventorycost,oritcanbetheamountofvaluealreadyaddedtothejob.Thefourpiecesofdatalistedabovearestaticdata,sincetheydonotdependontheschedule.Conversely,datathatarenotfixedinadvanceandthatdodependontheschedulearereferredtoasdynamicdata.Themostimportantdynamicdataarethefollowing:
222ManufacturingModelsStartingtimeSij.ThestartingtimeSijisthetimewhenjobjstartsitsprocessingonmachinei.Ifthesubscriptiisomitted,thenSjreferstothetimewhenjobjstartswithitsfirstprocessinginthesystem.CompletiontimeCij.ThecompletiontimeCijisthetimewhenjobjiscompletedonmachinei.Ifthesubscriptiisomitted,thenCjreferstothetimewhenjobjleavesthesystem.Animportantcharacteristicofaschedulingmodelisitsmachineconfig-uration.Thereareseveralimportantmachineconfigurations.Theremainingpartofthissectioncoversthemostbasicones.SingleMachineModels.Manyproductionsystemsgiverisetosinglemachinemodels.Forinstance,ifthereisasinglebottleneckinamulti-machineenvironment,thenthejobsequenceatthebottlenecktypicallydeterminestheperformanceoftheentiresystem.Insuchacase,itmakessensetoschedulethebottleneckfirstandallotheroperations(upstreamanddownstream)af-terward.Thisimpliesthattheoriginalproblemfirsthastobereducedtoasinglemachineschedulingproblem.Singlemachinemodelsarealsoimportantindecompositionmethods,whenschedulingproblemsinmorecomplicatedmachineenvironmentsarebrokendownintoanumberofsmallersinglema-chineschedulingproblems.Singlemachinemodelshavebeenthoroughlyanalyzedunderallkindsofconditionsandwithmanydifferentobjectivefunctions.Theresultisacollectionofrulesthat,whileeasytoidentifyandapply,oftenprovideoptimalsolutionsinthesinglemachineenvironment.Forexample,theEarliestDueDatefirst(EDD)rule,whichordersthejobsinincreasingorderoftheirduedates,hasbeenshowntominimizethemaximumlatenessamongalljobs.TheShortestProcessingTimefirst(SPT)rulehasbeenshowntominimizetheaveragenumberofjobswaitingforprocessing.ParallelMachineModels.Abankofmachinesinparallelisageneral-izationofthesinglemachinemodel.Manyproductionenvironmentsconsistofseveralstagesorworkcenters,eachwithanumberofmachinesinparal-lel.Themachinesataworkcentermaybeidentical,sothatajobcanbeprocessedonanyoneofthemachinesavailable.Parallelmachinemodelsareimportantforthesamereasonthatsinglemachinemodelsareimpor-tant:Ifoneparticularworkcenterisabottleneck,thenthescheduleatthatworkcenterwilldeterminetheperformanceoftheentiresystem.Thatbot-tleneckcanthenbemodelledasabankofparallelmachinesandanalyzedseparately.Attimes,themachinesinparallelmaynotbeexactlyidentical.Somemachinesmaybeolderandoperateatalowerspeed;or,somemachinesmaybebettermaintainedandcapableofdoinghigherqualitywork.Ifthatisthecase,thensomejobsmaybeprocessedonanyoneofthemmachines,whileotherjobsmaybeprocessedonlyonspecificsubsetsofthemmachines.Whenthe”machines”arepeople,thentheprocessingtimeofanoperationmaydependonthejobaswellasonthepersonoroperator.Oneoperator
2.2Jobs,Machines,andFacilities23Fig.2.1.Flexibleflowshopmayexcelinonetypeofjobwhileanotheroperatormaybemorespecializedinanothertype.FlowShopModels.Inmanymanufacturingandassemblysettings,jobshavetoundergomultipleoperationsonanumberofdifferentmachines.Iftheroutesofalljobsareidentical,i.e.,alljobsvisitthesamemachinesinthesamesequence,theenvironmentisreferredtoasaflowshop.Themachinesaresetupinseriesandwheneverajobcompletesitsprocessingononemachineitjoinsthequeueatthenext.Thejobsequencemayvaryfrommachinetomachine,sincejobsmayberesequencedinbetweenmachines.However,ifthereisamaterialhandlingsystemthattransportsthejobsfromonemachinetothenext,thenthesamejobsequenceismaintainedthroughoutthesystem.Ageneralizationoftheflowshopistheso-calledflexibleflowshop,whichconsistsofanumberofstagesinseries,withateachstageanumberofma-chinesinparallel.Ateachstageajobmaybeprocessedonanyoneofthemachinesinparallel(seeFigure2.1).Insome(flexible)flowshops,ajobmaybypassamachine(orstage)ifitdoesnotrequireanyprocessingthere,anditmaygoaheadofthejobsthatarebeingprocessedthereorthatarewaitingforprocessingthere.Other(flexible)flowshops,however,maynotallowbypass.JobShopModels.Inmulti-operationshops,jobsoftenhavedifferentroutes.Suchanenvironmentisreferredtoasajobshop,whichisageneral-izationofaflowshop(aflowshopisajobshopinwhicheachandeveryjobhasthesameroute).Thesimplestjobshopmodelsassumethatajobmaybeprocessedonaparticularmachineatmostonceonitsroutethroughthesystem(seeFigure2.2).Inothersajobmayvisitagivenmachineseveraltimesonitsroutethroughthesystem.Theseshopsaresaidtobesubjecttorecirculation,whichincreasesthecomplexityofthemodelconsiderably.Ageneralizationofthejobshopistheflexiblejobshopwithworkcentersthathavemultiplemachinesinparallel.Fromacombinatorialpointofview
242ManufacturingModelsFig.2.2.Jobshoptheflexiblejobshopwithrecirculationisoneofthemostcomplexmachineenvironments.Itisaverycommonsettinginthesemiconductorindustry.ThewaferfabdescribedinExample1.1.2isaclassicexampleofaflexiblejobshop;theroutesofthejobsareorder-specificandrequirerecirculation.SupplyChainModels.Moregeneralmodelsassumeaproductionen-vironmentthatconsistsofanetworkofinterconnectedfacilitieswitheachfacilitybeinga(flexible)flowshopora(flexible)jobshop.Insupplychainmanagementtheplanningandschedulingofsuchnetworksisveryimpor-tant.Thisplanningandschedulingmayfocusontheactualproductioninthevariousfacilitiesaswellasonthetransportationoftheproductswithinthenetwork.Intherealworldtherearemanymachineenvironmentsthataremorecomplicated.Nonetheless,thosedescribedabovearesofundamentalthattheiranalysesprovideinsightsthatareusefulfortheanalysesofmorecomplicatedenvironments.2.3ProcessingCharacteristicsandConstraintsJobprocessinghasmanydistinctivecharacteristicsandisoftensubjecttoconstraintsthatarepeculiar.Thissectiondescribessomeofthemostcommonprocessingcharacteristicsandconstraints.PrecedenceConstraints.Inschedulingproblemsajoboftencanstartonlyafteragivensetofotherjobshasbeencompleted.Suchconstraintsarereferredtoasprecedenceconstraintsandcanbedescribedbyaprecedenceconstraintsgraph.Aprecedenceconstraintsgraphmayhaveaspecificstruc-ture.Forexample,itmaytaketheformofasetofchains(seeFigure2.3.a),oratree(seeFigure2.3.b).InthesysteminstallationprojectofExample1.1.1someoftheactivitiesorjobsaresubjecttoprecedenceconstraints.MachineEligibilityConstraints.Inaparallelmachineenvironment,itmayoftenbethecasethatjobjcannotbeassignedtojustanyoneof
2.3ProcessingCharacteristicsandConstraints25(a) Chains(b) TreeFig.2.3.Precedenceconstraintsthemachinesavailable;itcanonlygoonamachinethatbelongstoaspecificsubsetMj.Asdescribedearlier,thismayoccurwhenthemmachinesinparallelarenotallexactlyidentical.Inthepapermillexample,thereareanumberofmachineswithdiffer-entcharacteristics,e.g.,dimensions,speeds,etc.Fromanoperationalpointofview,itisthenadvantageoustohavethelongrunsonthefastma-chines.WorkforceConstraints.Amachineoftenrequiresoneormorespecificoperatorstoprocessajobandafacilitymayhaveonlyafewpeoplewhocanoperatesuchamachine.Jobsthatneedprocessingonsuchamachinemayhavetowaituntiloneoftheseoperatorsbecomesavailable.Aworkforcemayconsistofvariouspools;eachpoolconsistsofoperatorswithaspecificskillset.ThenumberofoperatorsinpoollisdenotedbyWl.Iftheworkforceishomogeneous,i.e.,eachpersonhasthesamequalifications,thenthetotalnumberisdenotedbyW.Inaparallelmachineenvironmentjobshavetobescheduledinsuchawaythattheworkforceconstraintsaresatisfied.Becauseofsuchconstraints,machineschedulingandworkforceschedulingoftenhavetobedealtwithinanintegratedmanner.RoutingConstraints.Routingconstraintsspecifytherouteajobmustfollowthroughasystem,e.g.,aflowshoporjobshop.Agivenjobmayconsistofanumberofoperationsthathavetobeprocessedinagivensequenceororder.Theroutingconstraintsspecifytheorderinwhichajobmustvisitthevariousmachines.Routingconstraintsarecommoninmostmanufacturingenvironments.ConsideragainthewaferfabdescribedinExample1.1.2,inwhicheachorderorjobmustundergoanumberofdifferentoperationsatthevarious
262ManufacturingModelsstages.Anindividualjobthatdoesnotneedprocessingataparticularstagemaybeallowedtobypassthatstageandgoontothenext.Theinformationthatspecifiesthestagesajobmustvisitandthestagesitcanskipisembeddedintheroutingconstraints.MaterialHandlingConstraints.Modernassemblysystemsoftenhavematerialhandlingsystemsthatconveythejobsfromoneworkcentertoan-other.Thelevelofautomationofthematerialhandlingsystemdependsonthelevelofautomationoftheworkcenters.Iftheworkcentersarehighlyau-tomated(e.g.,roboticized),theprocessingtimesaredeterministicanddonotvary.Thematerialhandlingsystemmaythenhavetobeautomatedaswell.Whenmanualtasksareperformedattheworkcenters,thepaceofthema-terialhandlingsystemmaybeadjustablesincethepacemaydependontheprocessingtimesofthejobs.Amaterialhandlingsystemenforcesastrongdependencybetweenthestartingtimeofanygivenoperationandthecompletiontimesofitsprede-cessors.Moreover,thepresenceofamaterialhandlingsystemoftenlimitstheamountofbufferspace,whichinturnlimitstheamountofWork-In-Process.TheautomobileassemblyfacilitydescribedinExample1.1.3isapacedassemblylinewithamaterialhandlingsystemthatmovesthecars.Thecarsusuallybelongtoalimitednumberofdifferentfamilies.However,carsfromthesamefamilymaybedifferentbecauseofthevariousoptionpackagesthatareavailable.Thismaycauseacertainvariabilityintheprocessingtimesofthecarsatanygivenworkstation.Theprocessingtimesattheworkstationsaffecttheconfigurationoftheline,itspace,andthenumberofoperatorsassignedtoeachstation.SequenceDependentSetupTimesandCosts.Machinesoftenhavetobereconfiguredorcleanedbetweenjobs.Thisisknownasachangeoverorsetup.Ifthelengthofthesetupdependsonthejobjustcompletedandontheoneabouttobestarted,thenthesetuptimesaresequencedependent.Ifjobjisfollowedbyjobkonmachinei,thenthesetuptimeisdenotedbysijk.Forexample,paintoperationsoftenrequirechangeovers.Everytimeanewcolorisused,thepaintingdeviceshavetobecleaned.Thecleanuptimeoftendependsonthecolorjustusedandonthecolorabouttobeused.Inpracticeitispreferabletogofromlighttodarkcolors,becausethecleanupprocessistheneasier.Besidestakingvaluablemachinetime,setupsalsoinvolvecostsbecauseoflaborandwasteofrawmaterial.Forexample,machinesintheprocessandchemicalindustriesarenotstoppedwhengoingfromonegradeofmaterialtoanother.Instead,acertainamountofthematerialproducedatthestartofanewrunistypicallynotofacceptablequalityandisdiscardedorrecirculated.Ifjobjisfollowedbyjobkonmachineithenthesetupcostisdenotedbycijk.ConsiderthepapermachinesinExample1.1.4.Apapermachineproducesvarioustypesofpaper,characterizedbycolor,gradeandbasisweight.When
2.3ProcessingCharacteristicsandConstraints27themachineswitchesfromonecombinationofgrade,colorandbasisweighttoanother,acertainamountoftransitionproductgoestowaste.Themachineisnotproductiveforalengthoftimethatdependsonthecharacteristicsofthetwotypesofpaper.StorageSpaceandWaitingTimeConstraints.Inmanyproductionsystems,especiallythoseproducingbulkyitems,theamountofspaceavailableforWork-In-Process(WIP)storageislimited.Thisputsanupperboundonthenumberofjobswaitingforamachine.Inflowshopsthiscancauseblocking.Supposethestoragespace(buffer)betweentwosuccessivemachinesislimited.Whenthebufferisfull,theupstreammachinecannotreleaseajobthathasbeencompletedintothebuffer.Instead,thatjobhastoremainonthemachineonwhichithascompleteditsprocessingandthuspreventsthatmachinefromprocessinganotherjob.Make-to-StockandMake-to-Order.Amanufacturingfacilitymayopttokeepitemsinstockforwhichthereisasteadydemandandnoriskofobsolescence.ThisdecisiontoMake-To-Stockaffectstheschedulingprocess,becauseitemsthathavetobeproducedforinventorydonothavetightduedates.Whenthedemandratesarefixedandconstant,thedemandrateforitemsoftypejisdenotedbyDj.Inthecaseofsuchadeterministicdemandpro-cess,theproductionlotsizeisdeterminedbyatrade-offbetweensetupcostsandinventoryholdingcosts.Whenevertheinventorydropstozerothefacilityreplenishesitsstock.If,inthecaseofastochasticdemandprocess,theinven-torydropsbelowacertainlevel,thenthefacilityproducesmoreinordertoreplenishthein-stocksupply.Theinventorylevelthattriggerstheproductionprocessdependsontheuncertaintyinthedemandpattern,whiletheamountproduceddependsonthesetupcostsandinventoryholdingcosts.Make-to-Orderjobs,conversely,havespecificduedates,andtheamountproducedisdeterminedbythecustomer.ManyproductionfacilitiesoperatepartlyaccordingtoMake-to-StockandpartlyaccordingtoMake-to-Order.ConsidertheproductionplanofthepapermachineinExample1.1.4.Themillkeepsanumberofstandardproductsinstock,characterizedbytheircombinationofgrade,color,basisweightanddimensions.Themillhastocontroltheinventorylevelsofallthesecombinationsinsuchawaythatinventorycosts,setupcosts,andprobabilitiesofstockoutsareminimized.However,customersmayoccasionallyorderasizethatisnotkeptinstock,eventhoughthespecifiedgrade,colorandbasisweightsareproducedonaregularbasis.Thisitemisthenmadetoorder.Themillacceptstheorderthroughaproductionreservationsystemandinsertstheorderinitscyclicscheduleinsuchawaythatsetupcostsandsetuptimesaremini-mized.Preemptions.Sometimes,duringtheexecutionofajob,aneventoccursthatforcestheschedulertointerrupttheprocessingofthatjobinordertomakethemachineavailableforanotherjob.Thishappens,forinstance,when
282ManufacturingModelsarushorderwithahighpriorityentersthesystem.Thejobtakenoffthemachineissaidtobepreempted.Therearevariousformsofpreemptions.Accordingtooneform,thepro-cessingalreadydoneonthepreemptedjobisnotlost,andwhenthejobislaterputbackonthemachine,itresumesitsprocessingfromwhereitleftoff.Thisformofpreemptionisreferredtoaspreemptive-resume.Accordingtoanotherformofpreemption,theprocessingalreadydoneonthepreemptedjobislost.Thisformisreferredtoaspreemptive-repeat.TransportationConstraints.Ifmultiplemanufacturingfacilitiesarelinkedtooneanotherinanetwork,thentheplanningandschedulingofthesupplychainbecomesimportant.Thetransportationtimebetweenanytwofacilitiesaandbisknownanddenotedbyτmab(i.e.,thetimerequiredtomovetheproductsfromfacilityatofacilityb).Theremaybeconstraintsonthedeparturetimesofthetrucksaswellasonthequantitiesofgoodstobeshipped(i.e.,theremaybeupperboundsonthequantitiestobeshippedbecauseofthecapacitiesofthevehicles).Ofcourse,therestrictionsandconstraintsdescribedinthissectionarejustasampleofthosethatoccurinpractice.Therearemanyothertypesofprocessingcharacteristicsandconstraints.2.4PerformanceMeasuresandObjectivesManydifferenttypesofobjectivesareimportantinmanufacturingsettings.Inpractice,theoverallobjectiveisoftenacompositeofseveralbasicobjectives.Themostimportantofthesebasicobjectivesaredescribedbelow.ThroughputandMakespanObjectives.Inmanyfacilitiesmaximiz-ingthethroughputisoftheutmostimportanceandmanagersareoftenmea-suredbyhowwelltheydoso.Thethroughputofafacility,whichisequivalenttoitsoutputrate,isfrequentlydeterminedbythebottleneckmachines,i.e.,thosemachinesinthefacilitythathavethelowestcapacity.Thusmaximizingafacility’sthroughputrateisoftenequivalenttomaximizingthethroughputrateatthesebottlenecks.Thiscanbeachievedinanumberofways.First,theschedulermusttrytoensurethatabottleneckmachineisneveridle;thismayrequirehavingatalltimessomejobsinthequeuewaitingforthatmachine.Second,iftherearesequencedependentsetuptimessijkonthebottleneckmachine,thentheschedulerhastosequencethejobsinsuchawaythatthesumofthesetuptimes,or,equivalently,theaveragesetuptime,isminimized.Themakespanisimportantwhenthereareafinitenumberofjobs.ThemakespanisdenotedbyCmaxandisdefinedasthetimewhenthelastjobleavesthesystem,i.e.,Cmax=max(C1,…,Cn),whereCjisthecompletiontimeofjobj.Themakespanobjectiveiscloselyrelatedtothethroughputobjective.Forexample,minimizingthemakespan
2.4PerformanceMeasuresandObjectives29inaparallelmachineenvironmentwithsequencedependentsetuptimesforcestheschedulertobalancetheloadoverthevariousmachinesandminimizethesumofallthesetuptimes.Heuristicsthattendtominimizethemakespaninamachineenvironmentwithafinitenumberofjobsalsotendtomaximizethethroughputratewhenthereisaconstantflowofjobsovertime.ConsiderthepapermillinExample1.1.4andsupposethereareanumberofpapermachinesinparallel.Inthiscase,theschedulerhastwomainobjec-tives.Thefirstistoproperlybalancetheproductionoverthemachines.Thesecondistominimizethesumofthesequencedependentsetuptimesinordertomaximizethethroughputrate.DueDateRelatedObjectives.Thereareseveralimportantobjectivesthatarerelatedtoduedates.First,theschedulerisoftenconcernedwithminimizingthemaximumlateness.Joblatenessisdefinedasfollows.Letdjdenotetheduedateofjobj.ThelatenessofjobjisthenLj=Cj−dj(seeFigure2.4.a).ThemaximumlatenessisdefinedasLmax=max(L1,…,Ln).Minimizingthemaximumlatenessisinasenseequivalenttominimizingtheworstperformanceoftheschedule.Anotherimportantduedaterelatedobjectiveisthenumberoftardyjobs.Thisobjectivedoesnotfocusonhowtardyajobactuallyis,butonlyonwhetherornotitistardy.Thenumberoftardyjobsisastatisticthatiseasytotrackinadatabase,somanagersareoftenmeasuredbythepercentageofon-timeshipments.However,minimizingthenumberoftardyjobsmayresultinscheduleswithsomejobsverytardy,whichmaybeunacceptableinpractice.Aduedaterelatedobjectivethataddressesthisconcernisthetotaltardi-nessor,equivalently,theaveragetardiness.ThetardinessofjobjisdefinedasTj=max(Cj−dj,0)(seeFigure2.4.b)andtheobjectivefunctionisnj=1Tj.Supposedifferentjobscarrydifferentpriorityweights,wheretheweightofjobjiswj.Thelargertheweightofthejob,themoreimportantitis.Thenamoregeneralversionoftheobjectivefunctionisthetotalweightedtardinessnj=1wjTj.
302ManufacturingModelsT jd jC jd jC jL jd jC j(a) The lateness L j of job j (b) The tardiness T j of job j (c) Cost function in practicecostFig.2.4.Duedaterelatedobjectives
2.4PerformanceMeasuresandObjectives31ConsiderthesemiconductormanufacturingfacilitydescribedinExample1.1.2.Eachjob(or,equivalently,eachcustomerorder)hasaduedate,orcommittedshippingdate.Thejobsmayalsohavedifferentpriorityweights.Minimizingthetotalweightedtardinesswouldbeafairlysuitableobjectiveinsuchanenvironment,eventhoughaproductionmanagermaystillbecon-cernedwiththenumberoflatejobs.Noneoftheduedaterelatedobjectivesdescribedabovepenalizestheearlycompletionofajob.Inpractice,however,itisusuallynotadvantageoustocompleteajobearly,asthismayleadtostoragecostsandadditionalhandlingcosts.Costfunctions,inpractice,maylookliketheonedepictedinFigure2.4.c.SetupCosts.Itoftenpaystominimizethesetuptimeswhenthethrough-putratehastobemaximizedorthemakespanhastobeminimized.However,therearesituationswithinsignificantsetuptimessijkbutmajorsetupcostscsijk(ifsetupcostsaretheonlycostsinthemodel,thenthesuperscriptsmaybeomitted).Setupcostsarenotnecessarilyproportionalwithsetuptimes.Forexample,asetuptimeonamachinethathasamplecapacity(lotsofidletime)maynotbesignificant,eventhoughsuchasetupmaycausealargeamountofmaterialwaste.Work-In-ProcessInventoryCosts.AnotherimportantobjectiveistheminimizationoftheWork-In-Process(WIP)inventory.WIPtiesupcapital,andlargeamountsofitcanclogupoperations.WIPincreaseshandlingcosts,andolderWIPcaneasilybedamagedorbecomeobsolete.Productsareoftennotinspecteduntilaftertheyhavecompletedtheirpaththroughtheproduc-tionprocess.Ifadefectthatisdetectedduringfinalinspectioniscausedbyaproductionstepattheverybeginningoftheprocess,thenallWIPmaybeaffected.Thisisaveryimportantconsideration,especiallyinthesemicon-ductorindustrieswheretheyieldmaybeanywhereinbetween60and95%.Whenadefectisdetected,thecausehastobedeterminedimmediately.ThisthenpointstotheworkstationthatisresponsibleandgivesanindicationoftheproportionofdefectiveWIP.BecauseofsuchoccurrencesitpaystohavealowWIP.ThesekindsofconsiderationshaveledmanufacturingcompaniesinJapantotheJust-In-Time(JIT)concept.AperformancemeasurethatcanbeusedasasurrogateforWIPistheaveragethroughputtime.Thethroughputtimeisthetimeittakesajobtotraversethesystem.Minimizingtheaveragethroughputtime,givenacertainlevelofoutput,minimizesWIP.Minimizingtheaveragethroughputtimeisalsocloselyrelatedtominimizingthesumofthecompletiontimes,i.e.,nj=1Cj.Thislastobjectiveisequivalenttominimizingtheaveragenumberofjobsinthesystem.However,attimesitmaybeofmoreinteresttominimizethetotal
322ManufacturingModelsvaluethatistiedupasWIP.Ifthatisthecase,thenitismoreappropriatetominimizethesumoftheweightedcompletiontimes,i.e.,nj=1wjCj.FinishedGoodsInventoryCosts.Animportantobjectiveistomini-mizetheinventorycostsofthefinishedgoods.The(holding)costofkeepingoneitemoftypejininventoryforonetimeunitisdenotedbyhj.IftheproductionatafacilityisentirelyMake-To-Order,thenthefinishedgoodsinventorycostsareequivalenttoearlinesscosts.Conversely,ifafacility’smodeofproductionisMake-To-Stock,thenitcarriesinventoryonpurpose.Theproductionfrequencyofagivenitemandthelotsizedependonboththeinventorycarryingcostandthesetuptimeandcost.Inthisinstance,thedemandrateaswellastheuncertaintyinthedemanddeterminetheminimumsafetystockthathastobekept.Buteveninsuchacase,itisimportanttominimizethecostsofthefinishedgoodsinventory.TransportationCosts.Innetworksthatconsistofmanyfacilitiesthetransportationcostsmayrepresentasignificantpartofthetotalproductioncosts.Therearevariousmodesoftransportation,e.g.,truck,rail,air,andsea,andeachmodehasitsownsetofcharacteristicswithregardtospeed,costandreliability.Inpractice,thetransportationcostperunitisoftenincreasingconcaveinthequantitymoved.However,inthistextthecostoftransporting(moving)oneunitofproductfromfacilityatofacilitybisassumedtobeindependentofthequantitymovedandisdenotedbycmab.Therearemanyotherobjectivesbesidesthosementionedabove.Forex-ample,asJust-In-Time(JIT)conceptshavebecomemoreentrenchedinman-ufacturing,oneobjectivethatisincreasinginpopularityistheminimizationofthesumoftheearlinesses.InaJITsystem,ajobshouldnotbecompleteduntiljustbeforeitscommittedshippingdateinordertoavoidadditionalinventoryandhandlingcosts.Anothergoalforthescheduleristogenerateaschedulethatisasrobustaspossible.Ifascheduleisrobust,thenthenecessarychangesthathavetobemadeincaseofadisruption(e.g.,machinebreakdown,rushorder)tendtobeminimal.However,theconceptofrobustnesshasnotbeenwelldefinedyet,anditisalsonotclearhowrobustnessismaximized.Formoredetailsonthistopic,seeChapter15.2.5DiscussionFromtheprevioussectionsitisclearthatplanningandschedulingproblemsinmanufacturinghavemanydifferentaspects.InPartIIofthisbookweconsideronlysomeofthemostfundamentalaspects,butthereadershouldkeepinmindthattherearemanyotherimportantissues.
Exercises33Forexample,schedulingproblemsinpracticeareneverstatic,becausetheinputdatacontinuouslychange.Forexample,theweightofjobj,whichishardtomeasuretobeginwith,maynotbeconstant;itmaybetimedependent.Ajobthatisnotimportanttodaymaysuddenlybecomeimportanttomorrow.Planningandschedulingproblemsinpracticetypicallyhavemultipleob-jectives;theoverallgoalmaybeaweightedcombinationofseveralobjectives,e.g.,themaximizationofthetotalthroughputratecombinedwiththemini-mizationofthenumberoftardyjobs.Theweightsoftheindividualobjectivesmaychangefromdaytodayandmayevendependontheschedulerincharge.Also,personnelandoperationsschedulingproblemsarecloselyintertwined.Thisimpliesthatthetwoproblemsoftencannotbesolvedseparately;theymayhavetobesolvedtogether.Anotherpracticethatisquitecommonisjobsplitting.Ajobthatconsistsofacollectionorbatchofitemsmaybepartiallyprocessedononemachine,andpartiallyonanothermachine.Jobsplittingisaconceptthatismoregeneralthanpreemption,becauseajobcanbedividedintoseveralpartswhichmaybeprocessedconcurrentlyonvariousmachinesinparallel.Anotherformofjobsplittingcanoccurwhenajobisabatchofitemsthathastogoinaflowshopfromonemachinetothenext.Afterpartofabatchhasbeencompletedatonestageitcanstartitsprocessingonthenextmachinebeforetheentirebatchhasbeencompletedatthefirststage.Therearemanymoremachineconfigurationsthathavenotbeencoveredinthischapter.Forinstance,thereareenvironmentswhereeachjobneedsseveralmachinessimultaneously,dependinguponthesizeofthejob;or,jobsmayshareamachine.Anexampleisaportwithberthsforships;alargeshipmaytakethesamespaceastwosmallships,andsoon.Exercises2.1.Acontractorhasdecidedtouseprojectschedulingtechniquesfortheconstructionofabuilding.ThejobshehastodoarelistedinTable2.1.(a)Drawtheprecedenceconstraintsgraph.(b)Computethemakespanoftheproject.(c)Ifitwouldbepossibletoshortenoneofthejobsbyoneweek,whichjobshouldbeshortened?2.2.DefinetheearlinessofjobjasEj=max(dj−Cj,0).Inessencetheearlinessisthedualofthetardiness.Explainhowthemini-mizationofthesumoftheearlinessesrelatestotheJust-In-Timeconcept.2.3.Consideraflexibleflowshopenvironment.WheneverajobhastowaitbeforeitcanstartitsprocessingatthenextstageitisregardedasWork-In-Process(WIP),sinceitalreadyhasreceivedprocessingonmachinesupstream.
342ManufacturingModelsThecostsassociatedwithWIParestoragecosts,amountofvaluealreadyadded,costofcapital,andsoon.ExplainwhyhighWIPcostsmaypostponethestartingtimesofjobsoncertainmachines.2.4.ConsidertheCentralProcessingUnit(CPU)ofamainframecomputer.Jobswithdifferentpriorities(weights)comeinfromremoteterminals.TheCPUiscapableofprocessorsharing,i.e.,itcanprocessvariousjobssimulta-neously.(a)Explainhowtheprocessorsharingconceptcanbemodeledusingpre-emptions.(b)DescribeanappropriateobjectivefortheCPUtooptimize.2.5.Consideranenvironmentthatispronetomachinebreakdownsandsub-jecttohighinflationandfrequentstrikes.Whataretheeffectsofthesevari-ableson(a)theMaterialRequirementsPlanning,(b)theWork-In-Processlevels,and(c)thefinishedgoodsinventorylevels?2.6.Animportantaspectofascheduleisitsrobustness.Ifthereisarandomperturbationinarobustschedule(e.g.,machinebreakdown,unexpectedar-rivalofapriorityjob,etc.),thenthenecessarychangesinthescheduleareminimal.Thereisalwaysadesiretohaveaschedulethatisrobust.(a)Defineameasurefortherobustnessofaschedule.(b)Motivateyourdefinitionwithanumericalexample.JobDescriptionofJobDurationImmediatePredecessor(s)1Excavation4weeks–2Foundations2weeks13FloorJoists3weeks24ExteriorPlumbing3weeks15Floor2weeks3,46PowerOn1weeks27Walls10weeks58Wiring2weeks6,79CommunicationLines1weeks810InsidePlumbing5weeks711Windows2weeks1012Doors2weeks1013Sheetrock3weeks9,1014interiortrim5weeks12,1315exteriortrim4weeks1216Painting3weeks11,14,1517Carpeting1weeks1618Inspection1weeks17Table2.1.TableforExercise2.1
CommentsandReferences35CommentsandReferencesSomeofthebooksandarticlesmentionedinthecommentsandreferencesofChapter1presentfairlyelaborateproblemclassificationschemes(includingattimesaverydetailednotation)encompassingmanyoftheissuesdescribedinthischapter.Brucker,Drexl,M¨ohring,Neumann,andPesch(1999)presentacomprehensiveclassificationschemeforresourceconstrainedprojectscheduling.Conway,Maxwell,andMiller(1967)werethefirsttocomeupwithaclas-sificationschemeformachineschedulingandjobshopscheduling.RinnooyKan(1976),Lawler,LenstraandRinnooyKan(1982),Lawler,Lenstra,RinnooyKanandShmoys(1993),Herrmann,Lee,andSnowdon(1993),andPinedo(1995,2002,2008)developedmoreelaborateschemesthatcontainmanyofthemachineen-vironments,processingcharacteristicsandconstraints,andperformancemeasuresdiscussedinthischapter.MacCarthyandLiu(1993)developedaclassificationschemeforflexibleman-ufacturingsystems.Thisclassofsystemscontainstheclassofflexibleassemblysystemsasasubcategory.(Anothersubcategoryofflexiblemanufacturingsystemsistheclassofgeneralflexiblemachiningsystems(GFMS)whichisnotcoveredinthisbook;thissubcategorymayberegardedasjobshopswithautomatedmaterialhandlingsystems.)Someofthebooksthatcoverothermodelcategories,e.g.,lotscheduling,alsointroduceproblemclassificationschemes.However,theseotherschemeshavenotgainedmuchpopularityyet.
Chapter3ServiceModels3.1Introduction……………………………373.2ActivitiesandResourcesinServiceSettings….403.3OperationalCharacteristicsandConstraints….413.4PerformanceMeasuresandObjectives………443.5Discussion……………………………..463.1IntroductionServiceindustriesareinmanyaspectsdifferentfrommanufacturingindustries.Anumberofthesedifferencesaffecttheplanningandtheschedulingoftheactivitiesinvolved.Oneimportantdifferencecanbeattributedtothefactthatinmanufacturingitisusuallypossibletoinventorizegoods(e.g.,rawmaterial,Work-In-Process,andfinishedproducts),whereasinservicestherearetypicallynogoodstoinventorize.Thefactthatinmanufacturingajobcaneitherwaitorbecompletedearlyaffectsthestructureofthemodelsinamajorway.Inserviceindustries,ajobtendstobeanactivitythatinvolvesacustomerwhodoesnotliketowait.Planningandschedulinginserviceindustriesis,therefore,oftenmoreconcernedwithcapacitymanagementandyieldmanagement.Aseconddifferenceisbasedonthefactthatinmanufacturingthenumberofresources(whicharetypicallymachines)isusuallyfixed(atleastfortheshortterm),whereasinservicesthenumberofresources(e.g.,people,rooms,andtrucks)mayvaryovertime.Thisvariablemayevenbeapartoftheobjectivefunction.Athirddifferenceisduetothefactthatdenyingacustomeraserviceisamorecommonpracticethannotdeliveringaproducttoacustomerinamanufacturingsetting.Thisisoneofthereasonswhyrevenuemanagementplayssuchanimportantroleinserviceindustries.© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_3,37
383ServiceModelsThesedifferencesbetweenmanufacturingandservicesaffecttheprocessingrestrictionsandconstraintsaswellastheobjectives.Therefore,servicemodelstendtobeverydifferentfrommanufacturingmodels.Severalclassesofplanningandschedulingmodelsplayanimportantroleinserviceindustries.Oneclassofmodels,whichisimportantinbothmanufac-turingandservices,hasalreadybeendiscussedinthepreviouschapter.Thisclassconsistsoftheprojectplanningandschedulingmodels.Projectplanningandschedulingisnotonlyusefulinthebuildingofanaircraftcarrierbutalsointhemanagementofaconsultingproject.Eventhoughtheclassofprojectplanningandschedulingmodelsisimportantinserviceindustries,itisinthisbookonlydiscussedinthemanufacturingpart(PartII).Theremainderofthissectionprovidesashortdescriptionoffiveotherclassesofplanningandschedulingmodelsthatareimportantintheservicesindustries.Thefirstclassofmodelsconsideredintheservicespart(PartIII)consistsofmodelsforreservationsystemsandtimetabling.Thesetwosubclassesaremathematicallycloselyrelatedtooneanother;however,theyhavesimilaritiesaswellasdifferences.Inamodelforareservationsystemjobjhasadurationpjanditsstartingtimeandcompletiontimeareusuallyfixedinadvance,i.e.,thereisnoslack.Forexample,inacarrentalagencyajobisequivalenttothereservationofacarforagivenperiod.Theagencymaynotbeabletoprocessallthejobsthatpresentthemselves(i.e.,theymaynotbeabletoconfirmallreservations);itmayhavetodecidewhichjobstoprocessandwhichonesnot.Theobjectiveistypicallytoprocessasmanyjobsaspossible.Intimetabling(orrostering)jobjoractivityjmaybeameetingoranexamwithadurationpjwhichhastobescheduledwithinagiventimewindow.Theremaybeanearliestpossiblestartingtimerjandalatestpossiblecompletiontimedj,andtheremaybeacertainamountofslack.Intimetablingthestartingtimesandcompletiontimesofthejobsarenotfixedinadvance;but,inorderforanactivitytotakeplace,certainpeoplehavetobepresent.Thisimpliesthattwoactivitieswhichrequirethesamepersonoroperatortobepresentcannotbedoneatthesametime.Theremayalsoberestrictionsandconstraintswithregardtotheavailabilityoftheoperators(anoperatormayonlybeavailableduringagiventimeperiod).Oneobjectiveintimetablingmaybetheminimizationofthemakespansubjecttotheconstraintthatallactivitieshavetobescheduled.Anexampleofatimetablingproblemistheexamschedulingproblem;theexamsaresimilartojobsandtheroomsareequivalenttomachines.Twoexamsthathavetobetakenbythesamegroupofstudentsmaynotoverlapintime.Anotherexampleofatimetablingproblemistheschedulingofoperatingroomsinhospitals.Anoperationrequiresapatient,anoperatingroom,asurgeon,ananastaesiologistandothermembersofasurgicalteam.Theobjectiveistofindafeasibleschedulethatincludesalloperationsandsatisfiesallconstraints.Timetablinghassomesimilaritieswithworkforceschedulingaswell.Aworkforcemayconsistofvariouspoolsofdifferenttypesofoperators;apooloftypeconsistsofWoperatorsoftype;theintegerWisfixedovertime.
3.1Introduction39Thesecondclassofservicemodelsconsistsoftournamentschedulingmod-elsandbroadcasttelevisionschedulingmodels.Atournamentschedulingmodelinvolvesaleaguewithasetofteamsandafixednumberofgamesthathavetobeassignedtogiventimeslots.Thegamesmaybesubjecttomanyconstraints.Forexample,certaingameshavetobeplayedwithingiventimewindows.Thesequenceofgamesateamplaysinatournamentmaybesubjecttovariousconstraints;forexample,ateamshouldnotplaymorethantwoconsecutivegamesathomeormorethantwoconsecutivegamesaway.Atournamentschedulingproblemcanbecomparedtoaparallelma-chineschedulingprobleminwhichallthejobshavethesameprocessingtime.Moreover,thereareconstraintsthataresomewhatsimilartotheworkforceconstraintsinprojectscheduling,sinceateamcanplayatmostonegameinanygiventimeslot.Thethirdclassofservicemodelsarethetransportationschedulingmodels.Planningandschedulingisimportantforairlines,railroads,andshipping.Ajobmaybeatriporflightlegthathastobecoveredbyaship,plane,orvehicle.Aship,plane,orvehiclemayplayarolethatissimilartoamachine.Atriporflightlegjhastotakeplacewithinagiventimeframeandissubjecttogivenprocessingrestrictionsandconstraints.Theprocessingrestrictionsandconstraintsspecifyhowaparticulartriporflightlegcanbecombinedwithothertripsorflightlegsinafeasibleround-triplthatcanbeassignedtoaparticularvehicleandcrew.Roundtriplincursacostclandgeneratesaprofitπl.Theobjectiveiseithertominimizetotalcostormaximizetotalprofit.Thefourthclassofservicemodelsconcernsplanningandschedulinginhealthcareand,inparticular,inhospitals.Thereisaverylargevarietyofplanningandschedulingproblemsinhealthcare.Theyrangefromsurgeryschedulinginoperatingtheatresofhospitalstoappointmentschedulinginclinicsandradiationtreatmentcenters.Manyoftheseproblemshaverandomelements.Forexample,insurgeryschedulingthedurationsofthesurgeriesarerandomvariables.Intheschedulingofappointments,thereareprobabilitiesofno-shows.Anobjectivefunctionmayhavevariousdifferentcomponents:First,onemaywanttomaximizetheutilizationoftheresources(resourcesbeing,forexample,operatingrooms,radiationequipment,andsoon).Second,onemaywanttominimizetheexpectedwaitingtimeofpatientsandofsurgeons(forexample,asurgeonmayhavetowaitforanoperatingroomtobecomeavailable).Themathematicalmodelsformulatedfortheseproblemsarecloselyrelatedtoparallelmachineschedulingmodelsaswellastotimetablingmodelsandreservationsystems.Thefifthclassofservicemodelsconcernsworkforcescheduling,whichisaveryimportantaspectofplanningandschedulingintheserviceindustries.Workforceschedulingmodelstendtobequitedifferentfrommachineschedul-ingmodels.Workforceschedulingmayimplyeithershiftschedulinginaservicefacility(e.g.,acallcenter)orcrewschedulinginatransportationenvironment.Shiftschedulingmodelsaretheeasieronestoformulate:foreachtimeintervaltherearerequirementswithregardtothenumberofpersonnelthathaveto
403ServiceModelsbepresent.Timeintervalirequiresastaffingofbi(biinteger).Personnelcanbehiredfordifferentshiftsandthereisacostassociatedwitheachtypeofhire.Theobjectiveistominimizethetotalcost.Onecanarguethatshiftschedulingissomewhatsimilartomachinescheduling.Duringtimeintervali(whichmayhaveunitlength)bitaskshavetobedone,eachoneofunitlength.Theworkforceisinasenseequivalenttoanumberofmachinesinparallelwhichhavetoprocesstheunittasks.Atleastbiresourceshavetobeavailabletodotheseunittasksinintervali.Therearerestrictions,constraintsandcostsassociatedwithkeepingtheseresourcesavailableandtheobjectiveistominimizethetotalcost.Becauseofthespecialstructuresoftheshiftstheformulationofsuchaproblemturnsouttobedifferentfromtheformulationofatypicalmachineschedulingproblem.Theformulationsofcrewschedulingmodelstendtobedifferentfromtheformulationsofshiftschedulingmodelsaswell.Incontrasttotheshiftschedulingmodels,thecrewschedulingmodelsdonotassumeasteadystatepattern.Crewschedulingdependsverymuchonspecifictasksthathavetobedone.Crewschedulingarethereforeoftenintertwinedwithotherschedulingproceduresintheorganization(e.g.,theroutingandschedulingofplanesortrucks).Fromtheabove,itisclearthattherearesomesimilaritiesbetweenplan-ningandschedulingapplicationsinmanufacturingandplanningandschedul-ingapplicationsinservices.Theareasofintervalscheduling,timetablingandreservationmodelshavemanysimilaritieswithplanningandschedulinginmanufacturing.Reservationsystemsarecommonintransportation,healthcare,andhospitalityindustries;however,theyalsoplayaroleinmanufac-turingsettingswhencustomersmayreservespecifictimesongivenmachines.Planningandschedulingintransportationalsohavecommonalitieswithma-chinescheduling.Transportationmodelsareoftensimilartoparallelmachineschedulingmodels:aplaneissimilartoamachineandatriporflightlegissimilartoajobthathasaprocessingtime,releasedateandduedate;aturn-aroundtimeofaplaneissimilartoasequencedependentsetuptime.Inpractice,workforceschedulingmayoftenbeintertwinedwithotherschedulingfunctions.Forexample,machineschedulesmaydependonshiftschedulesandfleetschedulesmaydependoncrewschedules.However,inthisbookwedonotconsidermodelsthatintegrateworkforceschedulingwithotherschedulingfunctions.3.2ActivitiesandResourcesinServiceSettingsInmanufacturingajobusuallyrepresentsanactivitythattransformsaphys-icalcomponentandaddsvaluetoit.Inservicesanactivitytypicallyinvolvespeople.Fromtheprevioussectionitisclearthatanactivityinaservicesettingcantakemanyforms.Anactivitymaybe,forexample,(i)ameetingthathastobeattendedbycertainpeople,(ii)agamethathastobeplayedbetweentwoteams,
3.3OperationalCharacteristicsandConstraints41(iii)aflightlegthathastobecoveredbyaplane,(iv)anoperationthathastobedonebyasurgeononagivenday,or(v)apersonnelpositionthatmustbefilledinagiventimeperiod.Thecharacteristicsofajoboranactivityintheserviceindustriesareinsomeaspectssimilartothoseofajobinamanufacturingenvironment.Thedatainclude(i)aduration(processingtimepij),(ii)anearliestpossiblestartingtime(releasedaterj),(iii)alatestpossiblefinishingtime(duedatedj),(iv)aprioritylevel(weightwj).Theprioritylevelofanactivitydependsonitsprofitorcost.Anactivitymayhavemoreparameterswhichspecify,forexample,theadditionalresourcesthatmayberequiredinordertodotheactivity.Inmanufacturing,resourcesaretypicallyreferredtoasmachinesandtheconfigurationofthemachinesisgiven.Theymaybesetupinparallelorinseries,ortheproductmayfollowamorecomplicatedroute.Inservicemodels,theresourcesrequiredcantakemanydifferentforms.Aresourcecanbeaclassroom,meetingroom,hotelroom,stadium,rentalcar,oroperatingroom.Intransportation,itmaybeaplane,ship,truck,train,airportgate,dock,railroadtrack,person(e.g.,apilot),andsoon.Inahealthcaresetting,aresourcemaybeanoperatingroom,surgeon,anasthaesiologist,andsoon.Inserviceindustriesitisoftenimportanttosynchronizethetimingoftheuseofthedifferenttypesofresources.Itmaybethecasethat,inordertoperformacertainactivity,itisnecessarytohavetwoorthreedifferenttypesofresourcespresentatthesametime.Totransportagivencargofromonepointtoanother,atruck,adriveranddockingareashavetobeavailable.Ifatraingoesfromonestationtoanother,thenatrackbetweenthetwostationshastobefreewhilethetrainisonitsway.Eachtypeofresourcemayhavecharacteristicsorparametersthatareimportantfortheplanningandschedulingprocess.Forexample,ifresourceiisaroom,thenithasacapacityAi(numberofseats),acostci(orrevenue)perunittime,anavailabilityofcertainequipment(e.g.,audiovisual),andsoon.Aplane,shiportruckmayhaveacapacityAi,agivenspeedvi,andacertainrange.Ithasagivenoperatingcost,requiresacertaincrew,andcanbeusedonlyonspecificroutes.Ifaresourceisaperson,thenheorshemaybeaspecialist(e.g.,asurgeon).Aspecialistusuallyhasasetofcharacteristicsorskillsthatdeterminesthetypesofjobsheorshecanhandle.3.3OperationalCharacteristicsandConstraintsTheoperationalcharacteristicsandconstraintsinservicemodelsaremorediverseandoftenmoredifficulttospecifythanthoseinmanufacturingmodels.TimeWindows(ReleaseDatesandDueDates).Schedulingmeet-ings,exams,orgamesoftenmustsatisfytimewindowconstraints;suchtime
423ServiceModelswindowsareequivalenttoreleasedatesandduedatesinmanufacturing.Whenanairlineschedulesitsflights,ithastomakesurethateachflightdepartswithinitstimewindow.Atimewindowisoftendeterminedbyamarketingdepartmentthathasestimatesofpassengerdemandasafunctionofthede-parturetime.Thesameistruewithrespecttotraintimetabling.Eachtraininagivencorridorhasanidealarrivalanddeparturetimeforeachoneofthestationsinthecorridor.CapacityRequirementsandConstraints.Capacityrequirementsandconstraintsareimportantinreservationsystems,intimetablingofmeetings,aswellasintransportationplanningandscheduling.Ameetingorexammaybescheduledwithanumberofparticipantsinmind;thisnumberisanimportantparameterofthemeeting,sinceinafeasibleschedulethevalueofthisparametermustbelessthanthecapacityAiofthemeetingroom.Whenanairlineassignsitsplanes(whichmaybeofseveraldifferenttypes)tothevariousflightlegs,ithastotakeintoaccountthenumberofseatsoneachplane.Capacityrequirementsaresimilartothemachineeligibilityconstraintsinamanufacturingmodel.Inaparallelmachineenvironment,jobjmayoftennotbeprocessedonjustanyoneoftheavailablemachines,butratheronamachinethatbelongstoaspecificsubsetMj.AsdescribedinChapter2,thismaybethecasewhenthemmachinesarenotallexactlythesame.Capacityconstraintsarealsoimportantinrailwayscheduling.Thefactthatthereisusuallyonlyasingletrackineachdirectionbetweentwostationsmakesitimpossibleforonetraintopassanotherinbetweenstations.Railwayschedulingisparticularlydifficultbecauseofsuchtrackcapacityconstraints.Preemptions.Preemptionsarelessprevalentinservicesthaninmanu-facturing.Manytypesofserviceactivitiesareverydifficulttopreempt,e.g.,anoperationinahospital,aflightlegoragame.However,sometypesofservicejobscanbepreempted,butusuallysuchpreemptionsarenotallowedtooccuratjustanytime;preemptionsinservicestendtooccuratspecificpointsintimeratherthanatarbitrarypointsintime.Thetimeduringwhichameetingtakesplacemaybeacollectionofdisjointperiodsratherthanonecontiguousperiod.Ameetingmaybesplitupinsegmentsofonehourorseveralhourseach.SetupTimesandTurnaroundTimes.Resourcesmayrequiresetupsbetweenconsecutiveactivities.Rooms,planes,andtrucksmayhavetobecleanedandsetupforsubsequentmeetingsortrips.Thismayinvolveasetuptimesjkthatmayormaynotbesequencedependent.Setupsmayhaveeitherfixedorrandomdurations.Forexample,afteraplanearrivesatanairport,variousactivitieshavetobeperformedbeforeitcantakeoffagain.Arrivingpassengershavetobedeplaned,theplanehastoberefueledandcleaned,anddepartingpassengershavetoboardtheplane.Thisturnaroundtimeisequivalenttoasequencedependentsetuptime.
3.3OperationalCharacteristicsandConstraints43Inairportmodelsarunwaymaybeconsideredaresource;thetake-offsandlandingsaretheactivitiesthathavetobeperformedandtherearese-quencedependentsetuptimes.Inbetweentwoconsecutivetake-offsitmaybenecessarytokeeptherunwayidleforoneortwominutesinordertoallowairturbulencetodissipate.Thisidletimeislongerwhenasmallplanefollowsalargeplanethanviceversa.OperatorandToolingRequirements.Inanenvironmentwithparallelresources,activitiesmayhavetobescheduledinsuchawaythatadditionalrequirementsaremet(e.g.,specificoperatorsortoolshavetobeavailable)or,equivalently,theoperatorsand/ortoolshavetobeassignedinsuchawaythatallactivitiescanbedoneaccordingtotherules.Resourcesfrequentlyrequireoneormoreoperatorstodotheactivities.Operatorsorspecialistsmaybeofdifferenttypes,andofsometypestheremaybealimitedavailability.Theroutingandschedulingofplanesisthereforeoftenintertwinedwiththeschedulingofthecrews.Operatorandtoolingrequirementsplay,ofcourse,alsoaroleinmanufac-turingsettings.Amachineshopmayhaveonlyalimitednumberofoperatorswhocanworkonacertainmachine.Jobsthatneedprocessingonthatma-chinehavetowaituntiloneoftheseoperatorsbecomesavailable.Insuchcasesmachineschedulingandworkforceschedulingbecomeintertwined.IfthereisonlyonetypeofoperatorandthereareWofthem,thentheWoperatorsconstituteasingle,homogeneousworkforce.WorkforceSchedulingConstraints.Workforceschedulingandshiftassignmentsareusuallysubjecttomanyconstraints.Theyareoftenofaforminwhichpeopleworkagivennumberofconsecutivedays(e.g.,five)andthenhaveanumberofconsecutivedaysoff(e.g.,two).However,therearemanydifferenttypesofshiftsaswellasmanywaysofrotatingthem.InthehospitaldescribedinExample1.1.10thepersonnelrequirementsforweekendsdifferfromthoseforweekdays.Forinstance,onlyemergencyoperationsareperformedovertheweekend,whereasallplannedoperationsareperformedduringtheweek.Thisimpliesthattherearedifferentpersonnelrequirementsfortheoperatingroomsduringthesetwotimeperiods.Also,fewerbedsareoccupiedovertheweekendthanduringtheweek;thisaffectspersonnelrequirementsaswell.Unionrulesmayhaveasignificantimpactontheworkinghoursofperson-nel.Incallcenterseventhelunchbreaksandcoffeebreakshavetobesched-uledaccordingtorulesthatareapprovedbyaunion.WithregardtoairlinecrewstheFAAhasverytightregulationsconcerningthemaximumnumberofconsecutivehoursofflyingtimeandthemaximumnumberofhoursthatcanbeflownwithinagiventimeperiod.
443ServiceModels3.4PerformanceMeasuresandObjectivesSchedulingobjectivesinmanufacturingaretypicallyafunctionofthecomple-tiontimes,theduedates,andthedeadlinesofthejobs.Usually,thenumberofmachinesinamanufacturingenvironmentisfixed.Inserviceenvironmentstheobjectivesmaybesomewhatdifferent.Someobjectivesinservicesmayindeedalsobeafunctionofthecompletiontimes,theduedates,andthedeadlines;but,objectivesinservicesmayhaveanadditionaldimension.Incontrasttomanufacturing,thenumberofresourcesinaserviceenvironmentmaybevari-able(e.g.,thenumberoffull-timeandpart-timepeopleemployed).Becauseofthis,theremaybeadifferenttypeofobjectivethattriestominimizethenumberofresourcesusedand/orminimizethecostassociatedwiththeuseoftheseresources.Anobjectiveinaserviceenvironmentmaythusbeacombinationoftwotypesofobjectives:oneconcerningthetimingsoftheactivitiesandtheotherconcerningtheutilizationoftheresources.Ifthesetwotypesofobjectivesarecombinedintoasingleobjectivefunctionthenappropriateweightshavetobeprovided.Thatpartofaservicesobjectivethatissimilartoamanufacturingob-jectivetendstobetightlyconstrained:thetimewindowsinwhichactivitieshavetobescheduledareoftennarrowandmay,attimes,nothaveanyslackatall.Aproblemmayreducetoanassignmentprobleminwhichanactivity(meetingorgame)hastobeassignedtoagiventimeslotand/ortoagivenroomwiththeassignmenthavingacertaincost(orfit).Theobjectiveistominimizethetotalcostofalltheassignments.Intransportationscheduling,certainlegs(ortrips)havetobecombinedintofeasibleroundtripsthatsatisfyallconstraints.Theroundtripscreatedmustincludeallthelegsthathavetobecoveredataminimumcost.Makespan.ThemakespanCmaxisnotonlyimportantinprojectplanningandschedulingmodels,itisalsoimportantintimetablingandreservationsystems.Forexample,considerasetofexamsthathastobesqueezedintoagivenexamperiodofoneweek;aprimaryobjectivemaybetoschedulealltheexamswithinthatperiodandasecondaryobjectivemaybetominimizethenumberofroomsused.Sinceminimizingthemakespanisoftenequivalenttominimizingthesequencedependentsetuptimes,themakespanobjectivealsoplaysaroleintransportationsettingswhenturnaroundtimeshavetobeminimized.SetupCosts.Beforethetake-offofaplane,certainpreparationshavetobemadethatmaydependonthetypeofflight.Thepreparationsincludecleaning,catering,fueling,andsoon.Atothertimesmoreelaboratesetupsarerequired,whichmayincludeenginemaintenance,etc.Eachsetuphasacostthatmaybesequencedependent.
3.4PerformanceMeasuresandObjectives45costtimeIdeal departure timeFig.3.1.EarlinessandTardinessPenaltiesEarlinessandTardinessCosts;ConvenienceCostsandPenaltyCosts.Intransportationscheduling,thereareoften”ideal”departuretimes(determinedbythemarketingdepartment).Ifthedepartureofatrainorplaneisshiftedtoomuchfromitsidealdeparturetime,thenapenaltycostisincurred(duetoanexpectedlossofrevenue),seeFigure3.1.PersonnelCosts.Acostisassociatedwiththeassignmentofagivenpersontoaparticularshift.Thisbasiccostisusuallyknowninadvance.However,overtimemayberequired,theamountofwhichoftennotknowninadvance.Costofovertimeistypicallysignificantlyhigherthancostofregulartime.ConsiderthehospitalenvironmentinExample1.1.10,whichhasa24-hourstaffingrequirement.Iftheworkforceisunionized,thentheircollectivebargainingagreementsmaycontainstaffingrestrictionsandconstraints.Thescheduler’sobjectiveistodevelopavarietyofshiftpatternsandassignpeopletoshiftsinsuchawaythatallrequirementsaremet.Eachshiftpatternhasacostassociatedwithitandtheobjectiveistominimizetotalcost.Crewschedulinginatransportationenvironmenttendstobemorecom-plicatedthannursescheduling.Thenurseschedulingprobleminahospitalcanbesolvedmoreorlessindependentlyfromotheroptimizationproblemsinahospital.Thecrewschedulingproblemofanairline,ontheotherhand,isintertwinedwithotherproblemsthatmustbedealtwithbytheairline,e.g.,routeselection,fleetassignment,etc.
463ServiceModels3.5DiscussionItseemsthatplanningandschedulingresearchinthepasthasfocusedlessonservicesandmoreonmanufacturing(inspiteofthefactthattherearemanyplanningandschedulingapplicationsinservices).Planningandschedulingobjectivesinserviceindustriesareoftenconsiderablymorecomplicatedthanplanningandschedulingobjectivesinmanufacturing.Servicemodelsseem,therefore,hardertocategorizethanmanufacturingmodels.Inservices,itoftenoccursthatvariousplanningandschedulingprocesseshavetointeractwithoneanother;forexample,intheaviationindustryfleetschedulingandcrewschedulingarelinkedtooneanother,eventhoughtheobjectivefunctionsofthetwomodulesarequitedifferent.Infleetschedulingusuallythenumberofplanesutilizedhastobeminimized,whereasincrewschedulingthetotalcostofthecrewshastobeminimized.Otherschedulingmodulesintheaviationindustryarenotascloselylinkedtooneanother.Forexample,gateassignmentandrunwayschedulingatairportstendtobeindependentprocesses.Animportantareaofresearchnowadaysconcernsshiftschedulingincallcenters.Varioustypesofoperators,withdifferentskillsets,havetorespondtoallkindsofcalls.Eachtimeintervalrequiresanumberofoperators.Therearemultipleshiftpatternsandeachpatternhasitsowncost.Eachtypeofop-eratorineachtypeofshifthasitsowncoststructure.Shiftschedulingmodelsmayalsobecombinedwithmachineschedulingmodelsinmanufacturing.Planningandschedulinginmanufacturingandinservicesmayhavetodealwithmediumtermandlongtermissues,andalsowithshorttermandreactiveschedulingissues.Schedulinginservicestendstobemoreshorttermorientedandmorereactivethanschedulinginmanufacturing.Exercises3.1.Consideranairportterminalwithanumberofgates.Planesarriveanddepartaccordingtoafixedschedule.Uponarrivalaplanehastobeassignedtoagateinawaythatisconvenientforthepassengersaswellasfortheairportpersonnel.Certaingatesareonlycapableofhandlingnarrowbodyplanes.Modelthisproblemasamachineschedulingproblem.(a)Specifythemachinesandtheprocessingtimes.(b)Describetheprocessingrestrictionsandconstraints,ifany.(c)Formulateanappropriateobjectivefunction.3.2.Aconsultingcompanyhastoinstallabrandnewproductionplanningandschedulingsystemforaclient.Thisprojectrequiresthesuccesfulcompletionofanumberofactivities,namely
Exercises472497311156810Fig.3.2.PrecedenceconstraintsofasysteminstallationActivityDescriptionofActivityDuration1Installationofnewcomputerequipment8weeks2Testingofcomputerequipment5weeks3Developmentofthesoftware6weeks4Recruitingofadditionalsystemspeople3weeks5Manualtestingofsoftware2weeks6Trainingofnewpersonnel5weeks7Orientationofnewpersonnel2weeks8Systemtesting4weeks9Systemtraining7weeks10Finaldebugging4weeks11Systemchangeover9weeksTheprecedencerelationshipsbetweentheseactivitiesaredepictedinFigure3.2.(a)Computethemakespanoftheproject.(b)Ifitwouldbepossibletoshortenoneoftheactivitiesbyoneweek,whichactivityshouldbeshortened?3.3.Considerthereservationsysteminahotel.Peoplecallintomakereser-vations.Modeltheproblemasamachineschedulingproblem.(a)Whatarethemachines?(b)Areallthemachinesidentical?
483ServiceModels(c)Whatarethejobsandwhataretheirprocessingtimes?(d)Formulateanappropriateobjectiveforthehoteltooptimize.3.4.Consideranexamscheduleatalargeuniversity.Theexamshavetobescheduledinsuchawaythatpotentialconflictsforthestudentsareminimized.Thereareclassroomconstraints,becauseofthenumberofroomsavailableandbecauseoftheirrespectivesizes.(a)Howcanthisproblembemodeledasamachineschedulingproblem?(b)Whatarethemachines?(c)Arethereoperatorortoolingconstraints?Iftherearesuchconstraints,whataretheoperatorsandwhatarethetools?3.5.Inthedescriptionofaworkforceinthischapteritisassumedthataworkforcemayconsistofanumberofdifferentpoolsthatdonotoverlap.Eachpoolhasitsownparticularskillsetandtheskillsetsofthedifferentpoolsdonotoverlap.Giveaformalpresentationofaframeworkthatassumesafixednumberofskills,sayN.Eachworkerhasasubsetofallpossibleskills,However,theskillsetsoftwodifferentworkersmaypartiallyoverlap,i.e.,theymayhavesomeskillsincommonandeachworkermayhaveskill(s)thattheotheronedoesnothave.Howdoesthecomplexityofthisframeworkcomparetothecaseinwhichtheskillsetsdonotoverlap?3.6.Acompanyhasacentralwarehouseandanumberofclientsthathavetobesuppliedfromthatwarehouse.Thereisafleetoftrucksavailableandalldistancesareknown,i.e.,distancesbetweenclientsaswellasbetweenclientsandthewarehouse.Atruckcansupplyvariousclientsonasingleroute.However,thetotaldistanceatruckisallowedtocoverisboundedfromabove.Theobjectiveistominimizetotaldistancetraveledbyalltrucks.Showthatthisproblemisequivalenttoaparallelmachineschedulingproblemwithsetuptimes.CommentsandReferencesThenumberofbooksonplanningandschedulinginservicesisconsiderablysmallerthanthenumberofbooksonplanningandschedulinginmanufacturing.Modelsforreservationsystemsandtimetablingarefairlynewtopicsthatarecloselyrelatedtooneanother.Modelsforreservationsystems(alsoreferredtoasintervalschedulingmodels)havebeenconsideredinacoupleofchaptersinDemp-ster,LenstraandRinnooyKan(1982).Timetablinghasreceivedmoreattentionintheliterature.ThetextbookbyParker(1995)discussesthistopicandvariousproceedingsofconferencesontimetablinghaveappearedrecently,seeBurkeandRoss(1996),BurkeandCarter(1998),BurkeandErben(2001),BurkeandDeCausmaecker(2003),BurkeandTrick(2004),andBurkeandRudova(2006).Noframeworkhasyetbeenestablishedforplanningandschedulingmodelsintransportation.Aseriesofconferencesoncomputer-aidedschedulingofpublictrans-porthasresultedinanumberofveryinterestingproceedings,seeWrenandDaduna
CommentsandReferences49(1988),DesrochersandRousseau(1992),Daduna,Branco,andPintoPaixao(1995),Wilson(1999),andVossandDaduna(2001).Christiansen,FagerholtandRonen(2004)presentasurveyofshiproutingandscheduling.AvolumeeditedbyYu(1997)considersoperationsresearchapplicationsintheairlineindustry;thisvol-umecontainsseveralpapersonplanningandschedulingintheairlineindustry.Aconsiderableamountofworkhasbeendoneonplanningandschedulinginhealthcare.Brandeau,SainfortandPierskalla(2004)editedahandbookinOpera-tionsResearchandhealthcare;severalchaptersinthishandbookdealwithplanningandschedulingapplications.Personnelschedulinghasnotreceivedthesameamountofattentionasprojectschedulingorjobshopscheduling.However,thetextbyNandaandBrowne(1992)iscompletelydedicatedtopersonnelscheduling.Parker(1995)devotesasectiontostaffingproblems.Burke,DeCausmaecker,VandenBergheandVanLandeghem(2004)giveanoverviewofthestateoftheartofnurserostering.Severalofthebooksonplanningandschedulingintransportationhavechaptersoncrewscheduling.
PartIIPlanningandSchedulinginManufacturing4ProjectPlanningandScheduling………………….535MachineSchedulingandJobShopScheduling………..836SchedulingofFlexibleAssemblySystems……………1177EconomicLotScheduling………………………..1438PlanningandSchedulinginSupplyChains…………..173
Chapter4ProjectPlanningandScheduling4.1Introduction……………………………534.2CriticalPathMethod(CPM)………………564.3ProgramEvaluationandReviewTechnique(PERT)………………………………..604.4Time/CostTrade-Offs:LinearCosts………..634.5Time/CostTrade-Offs:NonlinearCosts……..704.6ProjectSchedulingwithWorkforceConstraints.714.7ROMAN:AProjectSchedulingSystemfortheNuclearPowerIndustry……………….744.8Discussion……………………………..784.1IntroductionThischapterfocusesontheplanningandschedulingofjobsthataresubjecttoprecedenceconstraints.Thesettingmayberegardedasaparallelmachineenvironmentwithanunlimitednumberofmachines.Thefactthatthejobsaresubjecttoprecedenceconstraintsimpliesthatajobcanstartwithitsprocessingonlywhenallitspredecessorshavebeencompleted.Theobjectiveistominimizethemakespanwhileadheringtotheprecedenceconstraints.Thistypeofproblemisreferredtoasaprojectplanningandschedulingproblem.Amoregeneralversionoftheprojectplanningandschedulingproblemassumesthattheprocessingtimesofthejobsarenotentirelyfixedinadvance.Aprojectmanagerhassomecontrolonthedurationsoftheprocessingtimesofthedifferentjobsthroughtheallocationofadditionalfundsfromabudgetthathehasathisdisposal.Sinceaprojectmayhaveadeadlineandacompletionafteritsdeadlinemayentailapenalty,theprojectmanagerhastoanalyzethetrade-offbetweenthecostsofcompletingtheprojectlateandthecostsofshorteningthedurationsoftheindividualjobs.© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,53DOI: 10.1007/978-1-4419-0910-7_4,
544ProjectPlanningandSchedulingAnothermoregeneralversionofthebasicprojectplanningandschedul-ingproblemassumesthatajob’sprocessingrequires,besidesamachine(ofwhichthereisalwaysoneavailable),alsovariousoperators.Thework-forcemayconsistofseveraldifferentpoolsofoperators;eachpoolhasafixednumberofoperatorswithaspecificskill.Becauseofworkforcelimi-tations,itmaysometimesoccurthattwojobscannotbeprocessedatthesametime,eventhoughbothareallowedtostartasfarastheprecedenceconstraintsareconcerned.Thistypeofproblemisinwhatfollowsreferredtoasprojectschedulingwithworkforceconstraints.(Intheliteraturethistypeofproblemisoftenreferredtoasresourceconstrainedprojectschedul-ing.)Thebasicprojectschedulingproblemwithoutworkforceconstraintsiseasyfromacomputationalpointofview.Optimalsolutionscanbefoundwithlittlecomputationaleffort.Ontheotherhand,projectschedulingwithworkforceconstraintsisveryhard.Thesetypesofplanningandschedulingproblemsoftenoccurinpracticewhenlargeprojectshavetobeundertaken.Examplesofsuchprojectsarerealestatedevelopments,constructionofpowergenerationcenters,softwarede-velopments,andlaunchingsofspacecraft.Otherapplicationsincludeprojectsinthedefenseindustry,suchasthedesign,developmentandconstructionofaircraftcarriersandnuclearsubmarines.Inmostoftheliteratureonprojectscheduling,ajobinaprojectisreferredtoasanactivity.However,inwhatfollowsweusethetermjobratherthanactivityinordertobeconsistentwiththeremainingchaptersinthispartofthebook.Theprecedencerelationshipsbetweenthejobsarethebasicconstraintsoftheprojectschedulingproblem.Therepresentationoftheprecedencecon-straintsasagraphmayfolloweitheroneoftwoformats.Oneformatisreferredtoasthe”job-on-arc”formatandtheotherasthe“job-on-node”format.Inthejob-on-arcformat,thearcsintheprecedencegraphrepresentthejobsandthenodesrepresentthemilestonesorepochs.Forexample,ifjobjisfollowedbyjobk,thentheprecedencegraphisasdepictedinFigure4.1.a.Thefirstnoderepresentsthestartingtimeofjobj,thesecondthecompletiontimeofjobjaswellasthestartingtimeofjobkandthethirdandlastnodethecompletiontimeofjobk.Inthejob-on-nodeformat,thenodesintheprecedencegraphrepresentthejobs,andtheconnectingarcstheprecedencerelationshipsbetweenthejobs.Ifjobjisfollowedbyjobk,thentheprecedencegraphtakestheformdepictedinFigure4.1.b.Althoughinpracticethefirstformatismorewidelyusedthanthesecond,thesecondhasanumberofadvantages.Adisadvantageofthejob-on-arcformatisanecessityforso-called“dummy”jobsthatareneededtoenforceprecedenceconstraintsthatotherwisewouldnothavebeenenforcable.Thefollowingexampleillustratestheuseofdummyjobs.
4.1Introduction55job jjob jjob kjob k(b) Job-on-node format(a) Job-on-arc formatFig.4.1.FormatsofprecedencegraphsExample4.1.1(SettingupaProductionFacility).Considertheprob-lemofsettingupamanufacturingfacilityforanewproduct.Theprojectconsistsofeightjobs.Thejobdescriptionsandthetimerequirementsareasfollows:JobDescriptionDuration(pj)1Designproductiontooling4weeks2Preparemanufacturingdrawings6weeks3Prepareproductionfacilityfornewtoolsandparts10weeks4Procuretooling12weeks5Procureproductionparts10weeks6Kitparts2weeks7Installtools4weeks8Testing2weeksTheprecedenceconstraintsarespecifiedbelow.ImmediateImmediateJobPredecessorsSuccessors1—42—53—6,7416,752663,4,5873,4886,7—
564ProjectPlanningandScheduling(a) Job-on-node format(b) Job-on-arc format14725381256478063Fig.4.2.PrecedencegraphsforExample4.1.1job jjob kFig.4.3.AdvantagesofdepictingnodesasrectanglesTheprecedencegraphispresentedinthejob-on-nodeformatinFigure4.2.aandinthejob-on-arcformatinFigure4.2.b.Fromthefigureitisclearthatthereisaneedforadummyjobinthejob-on-arcformat.Thenumberofdummyjobsrequiredtoenforcetheproperprecedencesinajob-on-arcrepresentationofalargeprojectcanbesubstantialandmayincreasethetotalnumberofjobsbyasmuchas10%.Anotheradvantageofthejob-on-nodeformatisthatnodesmaybede-pictedasrectanglesandthehorizontalsidesoftherectanglecanbeusedasatime-axisthatcorrespondstotheprocessingtimeofthejob.Ifjobkisallowedtostartafterhalfofjobjhasbeencompleted,thenthearcestablishingthisprecedencerelationshipcanemanatefromthemidpointofahorizontalsideoftherectangle,seeFigure4.3.Precedenceconstraintsareinwhatfollowspresentedinthejob-on-nodeformat.However,nodesaredepictedascirclesandnotasrectangles.4.2CriticalPathMethod(CPM)Considernjobssubjecttoprecedenceconstraints.Theprocessingtimeofjobjisfixedandequaltopj.Thereareanunlimitednumberofmachinesin
4.2CriticalPathMethod(CPM)57parallelandtheobjectiveistominimizethemakespan.Besidesamachine(andthereisalwaysoneavailable)ajobdoesnotrequireanyotherresource.Thealgorithmthatyieldsaschedulewithaminimummakespanisrela-tivelysimpleandcanbedescribedinwordsasfollows:Startattimezerowiththeprocessingofalljobsthathavenopredecessors.Everytimeajobcom-pletesitsprocessing,startprocessingthosejobsofwhichallthepredecessorshavebeencompleted.Inordertodescribethealgorithmmoreformallyweneedsomenotation.LetCjdenotetheearliestpossiblecompletiontimeofjobjandSjtheearliestpossiblestartingtime.Clearly,Cj=Sj+pj.Lettheset{allk→j}denotealljobsthatarepredecessorsofjobj.Thisimpliesthatifjobkisapredecessorofjobj,jobkhastobecompletedbeforejobjcanbestarted.Thefollowingalgorithmisreferredtoastheforwardprocedure.Itisbasedononesimplefact:ajobcanstartitsprocessingonlywhenallitspredeces-sorshavebeencompleted.Sotheearlieststartingtimeofajobequalsthemaximumoftheearliestcompletiontimesofallitspredecessors.Algorithm4.2.1(ForwardProcedure).Step1.Settimet=0.SetSj=0andCj=pjforeachjobjthathasnopredecessors.Step2.ComputeinductivelyforeachjobjSj=max{allk→j}Ck,Cj=Sj+pj.Step3.ThemakespanisCmax=max(C1,…,Cn).STOPItcanbeshownthatthisprocedureyieldsanoptimalscheduleandthemakespanofthisoptimalschedulecanbecomputedeasily.Intheresultingscheduleeachjobstartsitsprocessingatitsearliestpossiblestartingtimeandiscompletedatitsearliestpossiblecompletiontime.However,thismaynotbenecessaryinordertominimizethemakespan.Itmaybepossibletodelaythestartofsomeofthejobswithoutincreasingthemakespan.Thenextalgorithmisusedtodeterminethelatestpossiblestartingtimesandcompletiontimesofallthejobs,assumingthemakespaniskeptatitsminimum.Thisalgorithmisreferredtoasthebackwardprocedure.Theal-gorithmusestheCmax,whichisanoutputoftheforwardprocedure,asan
584ProjectPlanningandSchedulinginput.Inordertodescribethisalgorithmsomeadditionalnotationisneeded.LetCjdenotethelatestpossiblecompletiontimeofjobjandSjthelatestpossiblestartingtimeofjobj.Lettheset{j→allk}denotealljobsthataresuccessorsofjobj.Algorithm4.2.2(BackwardProcedure).Step1.Sett=Cmax.SetCj=CmaxandSj=Cmax−pjforeachjobjthathasnosuccessors,Step2.ComputeinductivelyforeachjobjCj=min{j→allk}Sk,Sj=Cj−pj.Step3.Verifythatmin(S1,…,Sn)=0.STOPSotheforwardproceduredeterminestheearliestpossiblestartingtimesandcompletiontimesaswellastheminimummakespan.Thebackwardproce-durestartsoutwiththeminimummakespanandcomputesthelateststartingtimesandlatestcompletiontimes,suchthattheminimummakespanstillcanbeachieved.Ajobofwhichtheearlieststartingtimeisearlierthanthelateststartingtimeisreferredtoasaslackjob.Thedifferencebetweenajob’slatestpossiblestartingtimeandearliestpossiblestartingtimeistheamountofslack,alsoreferredtoasfloat.Ajobofwhichtheearlieststartingtimeisequaltothelateststartingtimeisreferredtoasacriticaljob.Thesetofcriticaljobsformsoneormorecriticalpaths.Acriticalpathisachainofnon-slackjobs,beginningwithajobthatstartsattimezeroandendingwithajobthatcompletesitsprocessingatCmax.Theremaybemorethanonecriticalpathandcriticalpathsmaypartiallyoverlap.Example4.2.3(ApplicationoftheCriticalPathMethod).Consider14jobs.Theprocessingtimesaregivenbelow.Jobs1234567891011121314pj569127121061097875TheprecedenceconstraintsarepresentedinFigure4.4.Theearliestcomple-tiontimeCjofjobjcanbecomputedusingtheforwardprocedure.
4.2CriticalPathMethod(CPM)592497311156141281013Fig.4.4.PrecedencegraphforExample4.2.3Jobs1234567891011121314Cj511142321263332364243515056Thisimpliesthatthemakespanis56.Assumingthatthemakespanis56,thelatestpossiblecompletiontimesCjcanbecomputedusingthebackwardprocedure.Jobs1234567891011121314Cj512142430263436364343515156Thosejobsofwhichtheearliestpossiblecompletiontimesareequaltothelatestpossiblecompletiontimesarecriticalandconstitutethecriticalpath.Sothecriticalpathis1→3→6→9→11→12→14.Thecriticalpathinthiscasehappenstobeunique.Thejobsthatarenotonthecriticalpathareslack.TheCriticalPathMethodisrelatedtotheCriticalPath(CP)dispatchingrulethatisdescribedinAppendixC.AccordingtotheCPrule,wheneveramachineisfreed,thejobattheheadofthelongestchainisgiventhehighestpriority.Similarly,intheCPMmethodonehastomakesurethatthejobattheheadofthelongestchainisneverdelayed.However,sincethereareanunlimitednumberofmachinesandnoworkforceconstraintsintheenvironmentconsideredhere,otherjobsdonothavetobedelayedeither.Themethod,therefore,doesnotfunctionasapriorityrule.Inamoregeneralsettingwithalimitednumberofmachinesorwithworkforceconstraints,theCriticalPathrulemayactuallyhavetopostponethestartofjobsthatareattheheadofshorterchains.
604ProjectPlanningandScheduling4.3ProgramEvaluationandReviewTechnique(PERT)Incontrasttothesettingintheprevioussection,theprocessingtimesofthenjobsarenowrandomvariables.Themeanµjandthevarianceσ2jofeachoftheserandomvariablesareeitherknownorcanbeestimated.ThetechniquetodeterminetheexpectedmakespanoftheprojectisoftenreferredtoastheProgramEvaluationandReviewTechnique(PERT).ThealgorithmthatminimizestheexpectedmakespanisexactlythesameastheCriticalPathMethod.Ateachjobcompletion,alljobswhosepredecessorshaveallbeencompletedarestarted.However,thecomputationoftheexpectedmakespanisnowmorecomplicated.Weassumethatwehavethreepiecesofdatawithregardtheprocessingtimeofeachjob,namelypaj=theoptimisticprocessingtimeofjobj,pmj=themostlikelyprocessingtimeofjobj,pbj=thepessimisticprocessingtimeofjobj.Usingthesethreepiecesofdatatheexpectedprocessingtimeofjobjistypicallyestimatedbysettingˆµj=paj+4pmj+pbj6.BasedontheestimatesoftheexpectedprocessingtimesanestimatefortheexpectedmakespancanbeobtainedbyapplyingtheclassicalCriticalPathMethodwithfixedprocessingtimesthatareequaltotheestimatesfortheex-pectedprocessingtimes.ThisapplicationoftheCriticalPathMethodresultsinoneormorecriticalpaths.Anestimateoftheexpectedmakespanisthenobtainedbysummingtheestimatesfortheexpectedprocessingtimesofalljobsonacriticalpath.IfJcpdenotesthesetofjobsonacriticalpath,thenanestimatefortheexpectedmakespanisE(Cmax)=j∈Jcpˆµj.Toobtainsomefeelingforthedistributionofthemakespanintheoriginalproblem,oneproceedsasfollows.Computeanestimateforthevarianceoftheprocessingtimeofjobjbytakingˆσ2j=pbj−paj62.Inordertoobtainanestimateforthevarianceofthemakespanwefocusonlyonthecriticalpathanddisregardallotherjobs.Sincethejobsonthecriticalpathhavetobeprocessedoneafteranother,thevarianceofthetotalprocessingtimeofalljobsonthecriticalpathcanbeestimatedbytakingV(Cmax)=j∈Jcpˆσ2j.
4.3ProgramEvaluationandReviewTechnique(PERT)61Thedistributionofthemakespanisassumedtobenormal,i.e.,Gaussian,withmeanE(Cmax)andvarianceV(Cmax).Theseestimatesare,ofcourse,verycrude.Firstofall,theremaybemorethanonecriticalpath.Ifthereareseveralcriticalpaths,thentheactualmakespanisthemaximumofthetotalrealizedprocessingtimesofeachoneofthecriticalpaths.Sotheexpectedmakespanmustbelargerthantheestimatefortheexpectedmakespanobtainedbyconsideringasinglecriticalpath.Second,thetotalamountofprocessingonthecriticalpathisassumedtobenormallydistributed.Ifthenumberofjobsonthecriticalpathisverylarge,thenthisassumptionmaybereasonable(becauseoftheCentralLimitTheorem).However,ifthenumberofjobsonthecriticalpathissmall(say4or5),thenthedistributionofthetotalprocessingonthecriticalpathmaynotbethatclosetoNormal(seeExercise4.7).Example4.3.1(ApplicationofPERT).Considerthe14jobsofExample4.2.3.Thejobsaresubjecttothesameprecedenceconstraints(seeFigure4.4).However,nowtheprocessingtimesarerandom,withthefollowingPERTdata.Jobs1234567891011121314paj44810612451076672pmj568117121161087875pbj681418812127101581078Basedonthesedatathemeansandthevariancesoftheprocessingtimescanbeestimated.Jobs1234567891011121314ˆµj569127121061097875ˆσj0.330.6711.330.3301.330.3301.330.330.6601ˆσ2j0.110.4411.780.1101.780.1101.780.110.4401NotethattheestimatesofthemeansareequaltotheprocessingtimesusedinExample4.2.3.SothecriticalpathisthesamepathasinExample4.2.3,namely1→3→6→9→11→12→14.TheestimateofthemakespanisequaltothemakespaninExample4.2.3,i.e.,56.Ifwecomputetheestimateofthevarianceofthemakespan,weobtainV(Cmax)=j∈Jcpˆσ2j=2.66.Assumingthattheprojectdurationisnormallydistributedwiththeestimatedmeanandvarianceofthecriticalpath,thentheprobabilitythattheprojectiscompletedbytime60is
624ProjectPlanningandSchedulingΦ60−56√2.66=Φ2.449=0.993,whereΦ(x)denotestheprobabilitythatanormallydistributedrandomvari-ablewithmean0andvariance1islessthanx.Theprobabilitythatthisprojectwillbedonebytime60is99.3%andtheprobabilityitwillbecom-pletedaftertime60is0.7%.Notethatweignoredheretherandomnessinallthejobsthatarenotonthecriticalpath.Togetanideaoftheaccuracyoftheprobabilityofcompletingtheentireprojectbytime60,considerthepath1→2→4→7→10→12→14.Anestimateofthelengthofthispathis55.Thisimpliesthatthispathisonlyslightlyshorterthanthecriticalpath.Estimatingthevarianceofthispathyields7.33.Computingtheprobabilityoffinishingtheprojectbeforetime60byanalyzingthispathresultsinΦ60−55√7.33=Φ1.846=0.968.Accordingtothiscomputationtheprobabilitythattheprojectiscompletedaftertime60is3.2%.Thisprobabilityisclearlyhigherthantheoneobtainedbyconsideringthecriticalpath.Thereasonisclear:theestimateforthevarianceofthesecondpath(whichisnotcritical)issignificantlyhigherthantheestimateforthevarianceofthecriticalpath.ThisexamplehighlightsoneofthedrawbacksofthePERTprocedure.Thecriticalpath,whichisthelongestpath,mayhavearelativelylowvariance,whilethesecondlongestpath(inexpectation)mayhaveaveryhighvariance.Ifseveralsuchpaths,notnecessarilycriticalpaths,havehighvariances,thentheexpectedmakespanmaybesignificantlylargerthanthevaluecomputedusingthePERTmethod.Thenextexampleillustratesanextremecase.Example4.3.2(DrawbackofPERT).Considertheprecedencegraphde-pictedinFigure4.5.Thereareanumberofpathsinparallel.Onepathhasthelongesttotalexpectedprocessingtimebutitsvarianceiszero.Allotherpathshaveaslightlysmallertotalexpectedprocessingtimebutahighvariance.Theexpectedmakespanoftheprojectis,intheory,theexpectedmaximumofthek+1paths,i.e.,E(Cmax)=E(max(X1,X2,…,Xk+1)),whereXl,l=1,…,k+1,istherandomlengthofpathl.Considerthecasewherethelongestpathhasatotalprocessingtimeof51withzerovarianceandtheknoncriticalpathsallhavemean50andastandarddeviationof20.Assumethattheknoncriticalpathsdonotoverlapandaretherefore
4.4Time/CostTrade-Offs:LinearCosts63Fig.4.5.k+1pathsinparallelindependentofoneanother.IfthestandardPERTprocedureisused,thentheprobabilitythatthemakespanislongerthan60is0.However,iftheknoncriticalpathsaretakenintoconsideration,thenProb(Cmax>60)=1−Prob(Cmax≤60).Themakespanislessthan60onlyifallkpathsarelessthan60.Theproba-bilitythatoneofthepathswithmean50islessthan60isΦ60−5020=Φ0.5=0.691.SoProb(Cmax≤60)=(0.691)k.Ifkis5,thenProb(Cmax≥60)=0.84,i.e.,theprobabilitythatthemakespanislongerthan60is84%.4.4Time/CostTrade-Offs:LinearCostsThissectiongeneralizesthedeterministicframeworkpresentedinSection4.2.Inmanysituationsitmaybepossibletoshortentheprocessingtimeofajobbyallocatinganadditionalamountofmoneytothejob.Weassumeinthissectionandthenextthatthereisabudgetthatcanbeusedforallocatingadditionalfundstothevariousjobs.Weassumethattheprocessingtimeofajobisalinearfunctionoftheamountallocated.Thatis,themoremoneyallocated,theshorterthejob,asinFigure4.6.Thereisanabsoluteminimumprocessingtimepminjandanabsolutemaximumprocessingtimepmaxj.Thecostofprocessingjobjin
644ProjectPlanningandSchedulingp maxjp minjc bjc ajProcessing timeResourceallocatedFig.4.6.Relationshipbetweenjobprocessingtimeandresourceallocatedtheminimumamountoftimepminjiscajandthecostofprocessingjobjinthemaximumamountoftimepmaxjiscbj.Clearly,caj≥cbj.Letcjdenotethemarginalcostofreducingtheprocessingtimeofjobjbyonetimeunit,i.e.,cj=caj−cbjpmaxj−pminj.Sothecostofprocessingjobjinpjtimeunits,wherepminj≤pj≤pmaxj,iscbj+cj(pmaxj−pj).Assumethereisalsoafixedoverheadcostcothatisincurredonaperunittimebasis.Inordertodeterminetheminimumcostoftheentireprojectitisnecessarytodeterminethemostappropriateprocessingtimeforeachoneofthenjobs.Fromacomplexitypointofviewthisproblemiseasyandcanbesolvedinanumberofways.Wefirstpresentaveryeffectiveheuristicthatoftenleadstoanoptimalsolution(especiallywhentheproblemissmall).Therearetworeasonsforpresentingthisheuristic:first,itisusuallythewaytheproblemisdealtwithinpractice,and,second,itcanalsobeusedwhencostsarenonlin-ear.Aftertheheuristicwealsopresentalinearprogrammingformulationoftheproblem,thatalwaysresultsinanoptimalsolution.Beforedescribingtheheuristicitisnecessarytointroducesometerminol-ogy.Theprecedencegraphcanbeextendedbyincorporatingasourcenode(withzeroprocessingtime)thathasanarcemanatingtoeachnodethatrep-resentsajobwithoutpredecessors.Inthesamewayeachnodethatrepresentsajobwithoutsuccessorshasanarcemanatingtoasinglesinknode,whichalsohaszeroprocessingtime.Acriticalpathisalongestpathfromthesourcetothesink.LetGcpdenotethesubgraphthatconsistsofthecriticalpath(s)
4.4Time/CostTrade-Offs:LinearCosts65Cut setMinimal cut setSinkSourceFig.4.7.Cutsetsinthesubgraphofthecriticalpathsgiventhecurrentprocessingtimes.Acutsetinthissubgraphisasetofnodeswhoseremovalfromthesubgraphdisconnectsthesourcefromthesink.Acutsetissaidtobeminimalifputtinganynodebackinthegraphreestablishestheconnectionbetweenthesourceandthesink(seeFigure4.7).Thefollowingheuristictypicallyleadstoagood,butnotnecessarilyopti-malallocation.Algorithm4.4.1(Time/CostTrade-OffHeuristic).Step1.Setallprocessingtimesattheirmaximum.Determineallcriticalpath(s)withtheseprocessingtimes.ConstructthesubgraphGcpofthecriticalpaths.Step2.DetermineallminimumcutsetsinthecurrentGcp.Consideronlythoseminimumcutsetsofwhichallprocessingtimesarestrictlylargerthantheirminimum.IfthereisnosuchsetSTOP,otherwisegotoStep3.Step3.Foreachminimumcutsetcomputethecostofreducingallitsprocessingtimesbyonetimeunit.Taketheminimumcutsetwiththelowestcost.Ifthislowestcostislessthantheoverheadcostc0perunittimegotoStep4,otherwiseSTOP.
664ProjectPlanningandSchedulingStep4.Reducealltheprocessingtimesintheminimumcutsetbyonetimeunit.Determinethenewsetofcriticalpaths.RevisegraphGcpaccordinglyandgotoStep2.Inthisheuristictheprocessingtimesofthejobsarereducedineachit-erationbyonetimeunit.Itwouldbepossibletospeeduptheheuristicbymakingalargerreductionintheprocessingtimesineachiteration.Themax-imumamountbywhichtheprocessingtimesinaminimumcutsetcanbereducedisdeterminedbytwofactors.First,whilereducingtheprocessingtimesinaminimumcutsetoneofthesemayhititsminimum;whenthishap-pensthisparticularminimumcutsetisnotrelevantanymore.Second,whilereducingtheprocessingtimesanotherpathmaybecomecritical;whenthishappensthegraphGcphastoberevisedandanewcollectionofminimumcutsetshastobedetermined.Thereasonforpresentingtheheuristicintheformataboveisbasedonthefactthatitcanalsobeusedwhencostsarenonlinear(thenonlinearcaseisconsideredinthenextsection).Example4.4.2(ApplicationofTime/CostTrade-OffHeuristic).Con-sideragainamodificationofExample4.2.3.Theprocessingtimesintheorig-inalexamplearenowthemaximumprocessingtimes.Sothetotalprojecttime,i.e.,themakespanatthemaximumprocessingtimes,isequalto56.Theseprocessingtimescanbereducedtosomeextentatcertaincosts.Thefixedoverheadcostperunittime,co,is6.Sothetotaloverheadcostoverthemaximaldurationoftheprojectis336.Jobs1234567891011121314pmaxj569127121061097875pminj35795983764552caj2025201530403525302025352010cj72434344452248Job12constitutesaminimumcutsetwiththelowestcostofprocessingtimereduction.Reducingtheprocessingtimeofjob12from8to7costs2andsaves6inoverhead,resultinginanetsavingsof4.Withtheprocessingtimeofjob12equalto7,aparallelpathbecomescritical,namely11→13→14.SotherearetwosegmentsinGcp.Thefirstsegmentisthatpartoftheoriginalcriticalpathuptojob11;thesecondsegmentconsistsofthetwoparallelcriticalpathsfromjob11tojob14.Inthefirstsegmenttherearethreecutsets,eachoneconsistingofasinglejob.Reducingtheprocessingtimeofjob6byonetimeunitresultsin
4.4Time/CostTrade-Offs:LinearCosts67anetsavingsof3(theoverheadcostminusthemarginalcostofreducingtheprocessingtimeofjob6).Aftertheprocessingofjob6isreducedbyonetimeunit,thepath1→2→4→7→10→12becomescriticalaswell.ThereareanumberofcutsetsintheupdatedGcp.Onecutsetisformedbyjobs2and11.Thetotalcostofreducingtheprocessingtimesofthesetwojobsbyonetimeunitis4.Thenetsavingsistherefore6−4=2.(Job2hasnowhititsminimumprocessingtime.)Acutsetthatstillcanbeconsideredistheonethatconsistsofjobs4and11.Themakespancanbereducedbyshorteningtheprocessingtimesofjobs4and11byonetimeunit.Thetotalcostofthisreductionis5andthenetsavingsistherefore6−5=1.Bothjobs4and11canbereducedoneadditionaltimeunit,resultingoncemoreinanetsavingsof1.Job11hasnowhititsminimumprocessingtimeof4,andcannotbereducedfurther.Theresultingmakespanis51(seeFigure4.8.a).However,anumberofadditionalprocessingtimereductionscanbeaccom-plishedwithoutanincreaseintotalcosts.Forexample,jobs12and13canbothbereducedbytwotimeunitswithoutanincreaseincosts.Jobs4and6canalsobereducedbyonetimeuniteachwithoutanyincreaseincosts.Themakespancanthusbereducedto48withoutanyadditionalcost(seeFigure4.8.b).Thisimpliesthatthereareanumberofoptimalsolutions,allwithamakespanbetween48to51.Themakespancanbereducedevenfurther.However,furthertighteningincreasesthetotalcost.Althoughitusuallyprovidesagoodsolution,theheuristicisnotguar-anteedtoyieldanoptimalsolution.Thefollowingexampleillustratesacasewheretheheuristicyieldsasuboptimalsolution.Example4.4.3(ApplicationofTime/CostTrade-OffHeuristic).Con-sider5jobsthataresubjecttotheprecedenceconstraintspresentedinFigure4.9.Jobs12345pmaxj57843pminj46732cj62262Thefixedoverheadcostperunittime,co,is13.Iftheprocessingtimesofthejobsareattheirmaximum,themakespanis12andtherearethreecriticalpaths,seeFigure4.9.Thetotaloverheadforthedurationoftheprojectistherefore156.Theprocessingtimeofjobjcanbereducedatacostofcjperunittime.ApplyingAlgorithm4.4.1yieldsthefollowingresult.Inthebeginningtherearevariousminimumcutsets.Theminimumcutsetwiththesmallestcostincreaseconsistsofjobs3,5,and2.Reducingjobs3,5,and2onetimeunit
684ProjectPlanningandScheduling249731115614128101359101010997555564(b) Makespan Cmax = 482497311156141281013510101011997775564(a) Makespan Cmax = 51Fig.4.8.ProcessingtimesinExample4.4.2costs6,whichislessthanthegainof13achievedduetoreducedoverhead.Themakespanisnow11andthecriticalpathgraphremainsthesame.However,jobs2,3,and5arenowprocessedattheirminimumprocessingtimesandcannotbereducedfurther.Atthispointthereisonlyoneminimumcutsetremaining,namelyjobs1and4.Reducingbothjobs1and4byonetimeunitcosts12.Thisleadstoagainof1,sincetheoverheadperunittimeis13.Themakespanisnow10andalljobsareprocessedattheirminimumprocessingtimes.
4.4Time/CostTrade-Offs:LinearCosts69SinkSource12543Fig.4.9.PrecedencegraphforExample4.4.3Algorithm4.4.1,aspresented,stopsatthispoint.However,thesolutionobtainedisnotoptimal,sincejob5doesnothavetobeprocessedatitsmini-mumprocessingtime.Extendingitsprocessingtimetoitsoriginalprocessingtimedoesnotincreasethemakespanandreducesthetotalcost.Whenthecostfunctionsarelinear,asinExamples4.4.2and4.4.3,theoptimizationproblemcanbeformulatedasalinearprogram.Theprocessingtimeofjobj,pj,isadecisionvariableandnotaconstant.Theearliestpossiblestartingtimeofjobjisadecisionvariabledenotedbyxj,andthemakespanCmaxisalsoadecisionvariable.Thisimpliesatotalof2n+1variables.Theearlieststartingtimesofthejobswithoutanypredecessorsarezero.Thedurationofeachjobissubjecttotwoconstraints,namelypj≤pmaxjpj≥pminj.Thetotalcostoftheproject,asafunctionoftheprocessingtimesp1,…,pn,iscoCmax+nj=1cbj+cj(pmaxj−pj).Sincethefixedtermsinthiscostfunctiondonotplayaroleintheminimiza-tionoftheobjective,theobjectiveisequivalenttocoCmax−nj=1cjpj.IfAdenotesthesetofprecedenceconstraints,thentheproblemcanbefor-mulatedasfollows.
704ProjectPlanningandSchedulingminimizecoCmax−nj=1cjpjsubjecttoxk−pj−xj≥0forallj→k∈Apj≤pmaxjforalljpj≥pminjforalljxj≥0foralljCmax−xj−pj≥0foralljThislinearprogramhas2n+1decisionvariables,i.e.,p1,…,pn,x1,…,xnandCmax.Example4.4.4(LinearProgrammingFormulation).Considertheprob-lemdescribedinExample4.4.2.Thisproblemcanbeformulatedasalinearprogram.Thereare14jobs,sothereare2×14+1=29variables.Thevariablex1maybesetequalto0.Actually,thisvariabledoesnothavetobesetequaltozero;theoptimizationprocesswouldforcethisvariabletobezeroanyway.Theobjectivefunctionhas14+1=15terms.Thefirstsetofconstraintsconsistsof18constraints,oneforeacharcintheprecedencegraph.Inthemodelunderconsiderationallcostsarelinear.However,inpracticeitoftenoccursthatthereisaduedateassociatedwiththeentireproject,sayd.Ifthemakespanislargerthantheduedateapenaltyisincurredthatisproportionatewiththetardiness.Soinsteadofthelinearoverheadcostfunctionconsideredbefore,thereisapiecewiselinearcostfunction.Thealgorithmsandthelinearprogrammingformulationpresentedabovecanbeadaptedeasilytohandlethissituation.Thelinearprogrammingfor-mulationcanbemodifiedbyaddingtheconstraintCmax≥d.Itdoesnotmakesensetoreducethemakespantoavaluethatislessthanthecommittedduedate,sincethiswouldonlyincreasethecostandwouldnotresultinanybenefit.4.5Time/CostTrade-Offs:NonlinearCostsIntheprevioussectionthecostsofreducingtheprocessingtimesincreaselinearly.Inpractice,thecostsofreducingprocessingtimesaretypicallyin-creasingconvex(seeFigure4.10.a).Inthissectionweadoptadiscretetimeframework.Letcj(pj)denotethecostofprocessingjobjinpjtimeunits,pjbeinganinteger.Weassumethat
4.6ProjectSchedulingwithWorkforceConstraints71(a) Cost of reducing processing time (b) Overhead cost function co (t)Processing timeResourceallocatedOverheadcostfunctionco (t)timeFig.4.10.Nonlinearcoststheprocessingtimeofjobjmaytakeanyoneoftheintegervaluesbetweenpminjandpmaxj.Ifthecostisdecreasingconvexoverthisrange,thencj(pj−1)−cj(pj)≥cj(pj)−cj(pj+1).Theoverheadcostfunctioncomayalsobeafunctionoftime.Weassumethattheoverheadcostfunctionisincreasing(ornondecreasing)intime(seeFigure4.10.b).Letco(t)denotetheoverheadcostduringinterval[t−1,t].AneffectiveheuristicforthisproblemissimilartoAlgorithm4.4.1.Nowitisveryimportanttoreducetheprocessingtimesinacutsetineachiterationbyjustonetimeunit.Inthelinearcaseitwouldhavebeenpossibletohavealargerstepsizeinthereductionoftheprocessingtimes;inthiscaseitisnotadvisablesincetheshapeofthecostfunctionsmayaffectthestepsize.Asolutionforthisproblemisreachedeitherwhennominimumcutsetswithreducibleprocessingtimesremain,or,incasetherearesuchcutsets,themarginalcostofreducingsuchacutsetishigherthanthesavingsobtainedduetothereducedoverhead.Itisalsopossibletoformulatethisprobleminacontinuoustimeset-ting.Thisleadstoanonlinearprogrammingformulationthatisslightlymoregeneralthanthelinearprogrammingformulationdescribedintheprevioussection.Theobjectivenowisnonlinear,i.e.,Cmaxt=1co(t)+nj=1cj(pj).Theconstraintsarethesameasthoseinthelinearprogrammingformulationdescribedintheprevioussection.4.6ProjectSchedulingwithWorkforceConstraintsInmanyrealworldsettingstherearepersonnelorworkforceconstraints.Theworkforcemayconsistofvariousdifferentpoolsandeachpoolhasafixed
724ProjectPlanningandSchedulingnumberofoperatorswithaspecificskill.Eachjobrequiresforitsexecutionagivennumberfromeachpool.Iftheprocessingofsomejobsoverlapintime,thenthesumoftheirdemandsforoperatorsfromanygivenpoolmaynotexceedthetotalnumberinthatpool.Again,theobjectiveistominimizethemakespan.ThisproblemisreferredtoinwhatfollowsasProjectSchedulingwithWorkforceConstraints.(IntheliteraturethisproblemisoftenreferredtoasResourceConstrainedProjectScheduling.)Inordertoformulatethisproblemsomeadditionalnotationisneeded.LetNpdenotethenumberofdifferentpoolsintheworkforce.LetWdenotethetotalnumberofoperatorsinpoolandletWjdenotethenumberofoperatorsjobjneedsfrompool.Example4.6.1(WorkforceConstraints).Considerthefollowinginstancewithfivejobsandtwotypesofoperators,i.e.,Np=2.Therearefouroftype1andeightoftype2.Theprocessingtimesandworkforcerequirementsarepresentedinthetablebelow.Jobs12345pj84644W1j21312W2j30403Theprecedenceconstraintsarespecifiedbelow.ImmediateImmediateJobPredecessorsSuccessors1—42—53—541—52,3—Iftherewouldnothavebeenanyworkforceconstraints,thenthecriticalpathis1→4andCmax=12.However,thereisnofeasibleschedulewithCmax=12thatsatisfiestheworkforceconstraints.TheoptimalschedulehasCmax=18.Accordingtothisschedulejobs2and3areprocessedduringtheinterval[0,6],jobs1and5areprocessedduringtheinterval[6,14]andjob4isprocessedduringtheinterval[14,18].IncontrasttotheprojectschedulingproblemsdescribedinSections4.2and4.4,theprojectschedulingproblemwithworkforceconstraintsisintrinsi-callyveryhard.Thereisnolinearprogrammingformulationforthisproblem;however,thereisanintegerprogrammingformulation.Inordertoformulatethisproblemasanintegerprogram,assumethatallprocessingtimesarefixedandinteger.Introduceadummyjobn+1with
4.6ProjectSchedulingwithWorkforceConstraints73zeroprocessingtime.Jobn+1succeedsallotherjobs,i.e.,alljobswithoutsuccessorshaveanarcemanatingtojobn+1.Letxjtdenotea0−1variablethatassumesthevalue1ifjobjiscompletedexactlyattimetandthevalue0otherwise.Sothenumberofoperatorsjobjneedsfrompoolintheinterval[t−1,t]isWjt+pj−1u=txju.LetHdenoteanupperboundonthemakespan.Asimple,butnotverytight,boundcanbeobtainedbysettingH=nj=1pj.SothecompletiontimeofjobjcanbeexpressedasHt=1txjtandthemakespanasHt=1txn+1,t.Theintegerprogramcanbeformulatedasfollows:minimizeHt=1txn+1,tsubjecttoHt=1txjt+pk−Ht=1txkt≤0forj→k∈Anj=1Wjt+pj−1u=txju≤Wfor=1,…,Np;t=1,…,HHt=1xjt=1forj=1,…,nTheobjectiveoftheintegerprogramistominimizethemakespan.Thefirstsetofconstraintsensuresthattheprecedenceconstraintsareenforced,i.e.,ifjobjisfollowedbyjobk,thenthecompletiontimeofjobkhastobegreaterthanorequaltothecompletionofjobjpluspk.Thesecondsetofconstraintsensuresthatthetotaldemandforpoolattimetdoesnotsurpass
744ProjectPlanningandSchedulingtheavailabilityofpool.Thethirdsetofconstraintsensuresthateachjobisprocessed.Sincethisintegerprogramisveryhardtosolvewhenthenumberofjobsislargeandthetimehorizonislong,itistypicallysolvedviaheuristics.Itturnsoutthatevenspecialcasesofthisproblemarequitehard.However,foranumberofimportantspecialcasesheuristicshavebeendevelopedthathaveproventobequiteeffective.Thenextchapterfocusesonaveryimportantspecialcaseofthisprob-lem,namelythejobshopschedulingproblemwiththemakespanobjective.AveryeffectiveheuristicforthisparticularspecialcaseisdescribedinSec-tion5.4.OtherspecialcasesoftheprojectschedulingproblemwithworkforceconstraintsarethetimetablingproblemsdescribedinSections9.4and9.5.Severalheuristicsarepresentedthereaswell.4.7ROMAN:AProjectSchedulingSystemfortheNuclearPowerIndustryNuclearpowerplantshavetobeshutdownperiodicallyforrefueling,main-tenanceandrepair.Theseshutdownperiodsarereferredtoasoutages.Man-aginganoutageisadauntingprojectthatmayinvolveanywherefrom10,000to45,000jobsoractivities.Agoodscheduleisnotonlycrucialfornuclearsafetyreasonsbutalsoforeconomicreasons.Thecostperdayofshutdownisintheorderof$1,000,000.Theoutageschedulingproblemcanbedescribedasfollows:Givenasetofoutagejobs,resourcesandprecedenceconstraints,thesystemhastoassignresourcestojobsforspecifictimeperiodssothatthemakespanisminimizedandthejobsareperformedsafely.Therearefourtypesofjobsthathavetobedoneduringanoutage:(i)refuelingjobs,(ii)repairs,(iii)plantmodifications,and(iv)maintenancejobs.Anoutagemustbeplannedandmanagedtominimizeshutdownrisksbytak-ingappropriatepreventivemeasures.Themainsafetyfunctionsandsystemscomponentsthatmustbemonitoredduringanoutageare:(i)theACpowercontrolsystem,(ii)theprimaryandsecondarycontainment,(iii)thefuelpoolcoolingsystem,(iv)theinventorycontrol,(v)thereactivitycontrol,(vi)theshutdowncooling,and(vii)thevitalsupportsystems.
4.7ROMAN:AProjectSchedulingSystemfortheNuclearPowerIndustry75CurrentautomatedoutagemanagementapproachesarebasedonconventionalprojectschedulingtechniquessuchasCPMandPERT.Thesystemsusedinpracticearetypicallycommercialsoftwarepackagesthatarerathergeneric(seeAppendixE).Safetyandriskassessmentsareusuallydonemanually.Thisrequiresexperiencedpersonneltotakethedecisionswithregardtothevariousschedulingalternatives.ThefinalschedulesarevalidatedusingasimulatordevelopedbytheElectricPowerResearchInstitute(EPRI).GenericsystemsbasedonCPMandPERTtechniquessufferfromma-jorlimitationswithregardtothisspecificapplicationbecausetheygenerallycannottakesafetyconsiderationsintoaccount.TheRomeLaboratoryoftheU.S.AirForce,incollaborationwithEPRI,KamanScienceandKestrelInsi-tute,designedanddevelopedasystemspecificallyforschedulingoutages.ThesystemisreferredtoasROMAN(RomeLaboratoryOutageMAN-ager).Akeyconceptinenforcingsafetyconstraintsisthestateoftheplant,whichismeasuredincolors:green,yellow,orangeandred,inincreasingorderofrisk.Thestateoftheplantiscomputedfromcomplexdecisiontreesregardingsafetylevels;see,forexample,thesafetyfunctionoftheACpowercontroldepictedinFigure4.11.IfthereisajobbeingprocessedthatmaycauseACpowerloss,then,fortheplanttobeinayellowstate,twooff-siteACpowersourcesandthreeoperableemergencysafeguardbuseshavetobeavailable.ROMANusestwomeasuresassociatedwiththeprocessingtimeofajob:thedefiniteperiodandthepotentialperiod.Thedefiniteperiodofajobcorrespondstothatperiodoftimeduringwhichthejobisdefinitelybeingworkedon;itisthetimeintervalbetweenthelateststarttimeofthejob(LST)anditsearliestfinishtime(EFT).Thepotentialperiodofajobcorrespondstotheperiodoftimeduringwhichitmaybeprocessed,i.e.,thetimeperiodbetweenitsearlieststarttime(EST)anditslatestfinishtime(LFT).Inasense,thedefiniteperiodrepresentsalowerboundontheprocessingtimeofajobandthepotentialperiodanupperbound.Withregardtothesafetyconsiderations,twoconceptsaredefined,namely,thedefinitestateoftheplantandthepotentialstateoftheplant.Thedefinitestateoftheplantisassociatedwiththedefiniteperiodsandrepresentsthestateoftheplantwithregardtoagivensafetyfunction,assumingthatalljobsaredonewithintheirdefiniteperiods.Thepotentialstateoftheplantisassociatedwiththepotentialperiods.Theriskassociatedwiththepotentialstateoftheplantisalwayshigherthantheriskassociatedwiththedefinitestateoftheplant,sincetheprocessingtimesofthejobsinthepotentialstatearelongerthanthoseinthedefinitestate.TheschedulegenerationmethodadoptedinROMANisbasedonacon-straintprogrammingapproach,i.e.,aglobalsearchmethodcombinedwithconstraintpropagation(seeAppendixD).Initially,thesearchassumesthattheprocessingtimesareattheirminimum,i.e.,theprocessingtakesplaceduringthedefiniteperiods.Thesearchmethodattemptstogenerateafea-
764ProjectPlanningandSchedulingnoyes210210Operableemergencysafeguard busOperableemergencysafeguard busOperableemergencysafeguard busActivity withAC power losspotentialOff-sitesourcesavailableOperableemergencysafeguard busOperableemergencysafeguard busOperableemergencysafeguard busOff-sitesourcesavailable ⇒ 3 YELLOW 2 ORANGE ⇐ 1 RED4 YELLOW3 ORANGE⇐ 2 RED3 YELLOW2 ORANGE⇐ 1 RED4 YELLOW3 ORANGE⇐ 2 RED4 ORANGE⇐ 3 RED⇒ 3 GREEN2 YELLOW1 ORANGE0 REDFig.4.11.ExampleofadecisiontreeforthesafetyfunctionoftheACpowercontrolsiblescheduleinaconstructivemanner.Ateverystep,ajobthathasnotbeenscheduledyet(withagiventimewindow)isappendedtothepar-tialschedule.Thisisfollowedbyaconstraintpropagationprocedure,whichrecomputesthetimewindowsofalljobs,enforcingtheprecedencecon-straintsaswellasthesafetyconstraintsregardingthedefiniteperiods.Af-tertheconstraintpropagationstephasbeencompleted,theprocessrepeatsitself.Ifthescheduleobtainedbythisinitialglobalsearchdoesnotsatisfythesafetyrequirements,thetimewindowshavetobeadjustedinordertosatisfythesafetyconstraintsoverthepotentialperiodsofallthejobs.Thisphaseinvolvessolvinganotherglobalsearchproblem,thistimeconsideringthepo-tentialperiodsofthejobs.ROMANhasproventobesuccessfulsinceitextendsthefunctionalityof-feredbyexistingsoftwaretoolsforoutagemanagement.Allthetechnologicalconstraintscurrentlyusedforautomaticschedulegenerationareincorporatedintothesystem.Inaddition,ROMANgeneratesschedulesthatsatisfythesafetyconstraints.ThelatestversionofROMANschedulesupto2000jobsinapproximately1minuteonaSparc2.Theschedulesproducedareoftenbetterthanthoseobtainedmanuallysincemanynewpossibilitiesareexplored.
4.7ROMAN:AProjectSchedulingSystemfortheNuclearPowerIndustry77NAMORYTIVITCAemaN ytivitcATSETSLnoitaruDsrossecederP1VID SSOLPCA :stceffA 1-12D :emaNsrossecederPOCEP 08 :HSINIF 56 :TRATS 51 :NOITARUD 56 :TSL 56 :TSE)NAMOR( REGANAM EGATUO YROTAROBAL EMORtixE seitilitU strahC ttnaG nuR daoL sretemaraP011 001 09 08 07 06 05 04 03 02 01 00213-32D1-BRHR2-32D1-SUB12D1-CD4VID1-ARHR1-12DtixE seitilitU strahC ttnaG nuR daoL sretemaraPstcapmistcapmistcapmiTNALP-fO-ETATSELUDEHCS1VID SSOLPCA :stceffA 1-12D :emaNsrossecederPOCEP 08 :HSINIF 56 :TRATS 51 :NOITARUD 56 :TSL 56 :TSENAMORtixE seitilitU strahC ttnaG nuR daoL sretemaraPtixE seitilitU strahC ttnaG nuR daoL sretemaraP3-32D1-BRHR2-32D1-SUB12D1-CD4VID-ARHR11-12D011 001 09 08 07 06 05 04 03 02 01 01 SSER-VANU-MUN 0 ?REWOPCA :54- 13)1-42D 2-42D 3-SUB42D SSOLPCA( )1-42D 2-42D 3-SUB42D 2VID( PAM-SER-VANU)02US 01US 4VID 3VID 1VID( SSER-VA-TSILNAMORtixE seitilitU strahC ttnaG nuR daoL sretemaraPtixE seitilitU strahC ttnaG nuR daoL sretemaraPsutatS REWOP-CArewoP CA1VID2VID3VID4VID01US02US011 001 09 08 07 06 05 04 03 02 01 0Fig.4.12.StateofplantwithrespecttoACpowerHumanschedulerstendtoaggregatejobsandschedulethemasblocksratherthanexploreinterestingpossibilitiesthatoccurwhenthejobsarescheduledoneatatime.AkeyfeatureofROMANthatutilitypersonnelfindattractiveisthero-bustnessoftheschedulesthataregenerated.Thecurrentschedulergeneratesaschedulethatincludesstarttimewindowsforeachjob.Choosingthestarttimeofajobwithinitswindowallowsforafeasibleexecutionoftheschedule.Thewindowprovidesinformationabouthowcriticalthestarttimeofajobis;ifapredecessorjobisdelayed,ausercandecidewhethertherestillisenoughfreedominthestarttimewindowtoallowon-timecompletion,orwhetheritistimetoreschedulepartsoftheoveralloperation.ROMANcomesconfiguredwithaGraphicsUserInterface(GUI)thatdisplaysaninteractiveGanttchartforjobs,showingtheirstarttimewin-dow,duration,jobdescriptionandpredecessors.AnotherGanttchartshowsthehistoryofthestateoftheplantwithrespecttoACpower(seeFigure4.12).TheoutagemanagementproblemissomewhatsimilartotheprojectschedulingproblemwithworkforceconstraintsdiscussedinSection4.6.How-ever,itwouldhavebeendifficulttoformulatetheoutagemanagementproblemasanintegerprogramliketheonepresentedinSection4.6.Therearemanyconstraintsthatarenonlinear,subjective,andhardtoformulate.
784ProjectPlanningandScheduling4.8DiscussionProjectschedulingisprobablytheareainschedulingthathasreceivedthemostattentionfrompractitioners.Themoreimportantdeterministicprojectschedulingproblemsthatarenotsubjecttoworkforceconstraintsarewellsolved.Theproblemswithworkforceconstraintsareobviouslyharderthantheproblemswithoutworkforceconstraints.Theworkforceconstrainedprojectschedulingproblemisanimportantproblemintheschedulingliteraturesincemanyotherwell-knownproblemsarespecialcases.OneofthemostimportantspecialcasesofthisproblemisthejobshopschedulingproblemwiththemakespanobjectivethatisconsideredinChapter5.Theprojectschedulingproblemwithworkforceconstraintsbecomesajobshopschedulingproblemwheneachpoolconsistsofasingleoperatorwithaspecificskill.Anoperatoristhenequivalenttoamachineandtheprecedenceconstraintsintheprojectschedulingproblemareequivalenttotheroutingconstraintsinthejobshopschedulingproblem.Theworkforcestructuredescribedinthischapterissomewhatspecial.Alltheoperatorsinagivenpoolareassumedtohavethesameskillandbeabletodoataskthatcannotbedonebyanoperatorfromanyotherpool.Amoregeneralstructureisbasedonthefollowingassumptions:Theworkforcehasanumberofskillsets.Eachoperatorhasasubsetoftheseskillsets.Twooperatorsmayhaveskillsetsthatpartiallyoverlap,i.e.,thefirstoperatorhasskillsAandB,andthesecondoperatorskillsAandC.Whenoperatorshavepartiallyoverlappingskillsetstheprojectschedulingproblemwithworkforceconstraintsbecomesmorecomplicated.Whentheprocessingtimesarerandom,thentheproblemsbecomeevenharder.Evenwithoutworkforceconstraints,problemswithrandomprocess-ingtimesarehard;thePERTproceduremaynotalwaysgiveasatisfactorysolution.Workforceconstrainedprojectschedulingproblemswithrandompro-cessingtimeshavenotreceivedmuchattentionintheliterature.Thisisanareathatclearlydeservesmoreattention.Exercises4.1.Considerthejob-on-noderepresentationdepictedinExample4.2.3.Transformthisjob-on-noderepresentationintoajob-on-arcrepresentation.Comparethenumberofnodesandarcsinthefirstrepresentationwiththenumberofnodesandarcsinthesecondrepresentation.Canyoumakeanygeneralstatementwithregardtosuchacomparison?4.2.Constructthesmallestprecedenceconstraintsgraphinajob-on-nodeformatthatrequiresadummyjobinthecorrespondingjob-on-arcformat(Hint:Youneed4jobs).
Exercises794.3.ConsiderthesetupoftheproductionfacilitydescribedinExample4.1.1.Thedurationsofthe8jobsaretabulatedbelow.Jobs12345678pj46101210242(a)Computethemakespananddeterminethecriticalpath(s).(b)Supposethatthedurationofjob7canbereducedby3weeksto1week.Computethenewmakespananddeterminethenewcriticalpath(s).4.4.Considertheinstallationoftheproductionplanningandschedulingsys-temdescribedinExercise3.1.Jobs1234567891011pj85632524749(a)Computethemakespananddeterminethecriticalpath.Supposethateachjobcanbeshortenedatacertainexpense.Theoverheadcostis6perweek(intensofthousandsofdollars).Thecostfunctionsarelinear.Theminimumandmaximumprocessingtimesaswellasthemarginalcostsaretabulatedbelow.Jobs1234567891011pmaxj85632524749pminj53422323537caj3025201530403525302030cj72212344454(b)ApplyAlgorithm4.4.1tothisinstance.(c)Verifywhetherthesolutionobtainedin(b)isoptimal.4.5.ConsiderExample4.2.3.Assumethatthereareanunlimitednumberofmachinesinparallelandeachmachinecandoanyoneofthe14jobs.(a)Whatistheminimumnumberofmachinesneededtoachievethemin-imummakespan?Callthisminimumnumbermmin.(b)Ifthenumberofavailablemachinesismmin−1,byhowmuchdoesthemakespangoup?(c)Plottherelationshipbetweenthemakespanandthenumberofma-chinesavailable,m=1,…,mmin.4.6.ConsideragainExample4.1.1.Supposeitispossibletoaddresourcestothevariousjobsonthecriticalpathsinordertoreducethemakespan.Theoverheadcostcoperunittimeis6.Thecostsarelinearandthemarginalcostsaretabulatedbelow.
804ProjectPlanningandSchedulingJobs12345678pmaxj46101210242pminj257108121caj2025201530403525cj43423244(a)Determinetheprocessingtimesandthemakespanofthesolutionwiththetotalminimumcost.Isthesolutionunique?(b)Plottherelationshipbetweenthetotalcostoftheprojectandthemakespan.4.7.Considertwoindependentrandomvariableswiththefollowingthreepointdistribution.Prob(X=1)=0.25Prob(X=2)=0.50Prob(X=3)=0.25Considertheconvolutionorsumofthetwoindependentrandomvariableswiththisthreepointdistribution.Thisnewdistributionhasmassonthepoints2,3,4,5,and6.(a)Computetheexpectationandthevarianceofthisdistribution.(b)Plotthedistributionfunctionandplotthedistributionfunctionofthenormaldistributionwiththesamemeanandvariance.(c)Dothesamewiththeconvolutionofthreeindependentrandomvari-ableswiththeoriginalthreepointdistribution.4.8.ConsiderthefollowingPERTversionoftheinstallationoftheproductionplanningandschedulingsystemdescribedinExercises3.1and4.4.Jobs1234567891011paj21521411618pmj85632524749pbj1497436378710(a)Rankthepathsaccordingtothemeansoftheirtotalprocessingtime.(b)Rankthepathsaccordingtothevarianceintheirtotalprocessingtimes.Whichpathhasthehighestvariance?Whichpathhasthelowestvari-ance?(c)Computetheprobabilitythatthemakespanislargerthan27followingthestandardPERTprocedure.(d)Computetheprobabilitythatthemakespanislargerthan27bycon-sideringonlythepathwiththelargestvariance.4.9.FormulatetheresourceallocationproblemdescribedinExample4.4.3asalinearprogram.
CommentsandReferences814.10.Formulatethefollowingprojectschedulingproblemwithworkforcecon-straintsasanintegerprogram.Jobs123pj462W1j231W2j340Bothworkforcepoolshavefouroperators.Theprecedenceconstraintsarespecifiedbelow.ImmediateImmediateJobPredecessorsSuccessors1––2–332–CommentsandReferencesTheCriticalPathMethodwasfirstpresentedinanindustryreportofDuPontandRemingtonRandbyWalkerandSayer(1959)andthePERTmethodwasfirstdescribedinaU.S.NavyreportunderthetitlePERT(1958).Sincethenthesetwosubjectshavebeenthefocusofmanybooks;see,forexam-ple,ModerandPhilips(1970),WiestandLevy(1977),Kerzner(1994),Neumann,Schwindt,andZimmermann(2001),andDemeulemeesterandHerroelen(2002).Projectschedulingisoftenalsocoveredinbroaderbasedschedulingandproductionplanningandcontrolbooks;see,forexample,Baker(1974),andMortonandPentico(1993).PERThasreceivedasignificantamountofattentionfromtheresearchcommu-nityaswell;see,forexample,Fulkerson(1962),Elmaghraby(1967),andSasieni(1986).Projectschedulingwithworkforceconstraintsisintheliteratureusuallyre-ferredtoasresourceconstrainedscheduling.Thisareaalsohasreceivedanenormousamountofattentionfromtheresearchcommunity;see,forexample,Balas(1970),Talbot(1982),Patterson(1984),Kolisch(1995),andBrucker,Drexl,M¨ohring,Neu-mann,andPesch(1999).TheROMANsystemisdescribedindetailbyAlguireandGomes(1996)andGomes,Smith,andWestfold(1996).
Chapter5MachineSchedulingandJobShopScheduling5.1Introduction……………………………835.2SingleMachineandParallelMachineModels…845.3JobShopsandMathematicalProgramming….865.4JobShopsandtheShiftingBottleneckHeuristic895.5JobShopsandConstraintProgramming…….955.6LEKIN:AGenericJobShopSchedulingSystem1045.7Discussion……………………………..1115.1IntroductionThischapterfocusesonjobshops.Therearenjobsandeachjobvisitsanumberofmachinesfollowingapredeterminedroute.Insomemodelsajobmayvisitanygivenmachineatmostonceandinothermodelsajobmayvisiteachmachinemorethanonce.Inthelattercaseitissaidthatthejobshopissubjecttorecirculation.Ageneralizationofthebasicjobshopisaso-calledflexiblejobshop.Aflexiblejobshopconsistsofacollectionofworkcentersandeachworkcenterconsistsofanumberofidenticalmachinesinparallel.Eachjobfollowsapredeterminedroutevisitinganumberofworkcenters;whenajobvisitsaworkcenter,itmaybeprocessedonanyoneofthemachinesatthatworkcenter.Jobshopsareprevalentinindustrieswhereeachcustomerorderisuniqueandhasitsownparameters.Waferfabsinthesemiconductorindustryoftenfunctionasjobshops;anorderusuallyimpliesabatchofacertaintypeofitemandthebatchhastogothroughthefacilityfollowingacertainroutewithgivenprocessingtimes.Anotherclassicalexampleofajobshopisahospital.Thepatientsinahospitalarethejobs.Eachpatienthastofollowagivenrouteandhastobetreatedatanumberofdifferentstationswhilegoingthroughthesystem.© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_5,83
845MachineSchedulingandJobShopSchedulingSomeofthejobshopproblemsconsideredinthischapterare,fromamathematicalpointofview,specialcasesoftheprojectschedulingproblemwithworkforceconstraintsdescribedinSection4.6.ThesejobshopproblemsareNP-hardandcannotbeformulatedaslinearprograms.However,theycanbeformulatedeitherasintegerprogramsorasdisjunctiveprograms.Thischapterisorganizedasfollows.Thesecondsectionfocusesonjobshopsthatconsistofasingleworkcenter;thesejobshopsarebasicallyequiv-alenttoeitherasinglemachineoranumberofmachinesinparallel.Variousobjectivefunctionsareconsidered.Thethirdsectionpresentsamathemat-icalprogrammingformulationofajobshopwithasinglemachineateachworkcenterandthemakespanasobjective.Thefourthsectiondescribesthewell-knownshiftingbottleneckheuristicwhichistailor-madeforthejobshopandcanbeadaptedtotheflexiblejobshopaswell.Thefifthsectionfocusesonanentirelydifferentapproachforthejobshopwiththemakespanobjec-tive,namelytheconstraintprogrammingapproach.Thisapproachhasgainedaconsiderablepopularityoverthelastdecade.Thesubsequentsectionde-scribestheLEKINschedulingsystemwhichisspecificallydesignedforjobshopsandflexiblejobshops.Thediscussionsectionlistssomeotherpopularsolutiontechniquesthathavebeenusedinpracticeforjobshopscheduling.Allthemodelsconsideredinthischapterassumethatjobsarenotallowedtobepreempted.Ajob,oncestartedonamachine,remainsonthatmachineuntilitiscompleted.5.2SingleMachineandParallelMachineModelsAsinglemachineisthesimplestcaseofajobshopandaparallelmachineenvironmentisequivalenttoaflexiblejobshopthatconsistsofasinglework-center.Decompositiontechniquesformorecomplicatedjobshopsoftenhavetoconsidersinglemachineandparallelmachinesubproblemswithintheirframework.Considerasinglemachineandnjobs.Jobjhasaprocessingtimepj,areleasedaterjandaduedatedj.Ifrj=0anddj=∞,thentheprocessingofjobjisbasicallyunconstrained.ItisclearthatthemakespanCmaxinasinglemachineenvironmentdoesnotdependontheschedule.Forvariousotherobjectivescertainpriorityrulesgenerateoptimalschedules.Iftheobjectivetobeminimizedisthetotalweightedcompletiontime,i.e.,wjCj,andtheprocessingofthejobsisunconstrained,thentheWeightedShortestProcessingTimefirst(WSPT)rule,whichschedulesthejobsindecreasingorderofwj/pj,isoptimal.IftheobjectiveisthemaximumlatenessLmaxandthejobsareallreleasedattime0,thentheEarliestDueDatefirst(EDD)rule,whichschedulesthejobsinincreasingorderofdj,isoptimal.BoththeWSPTruleandtheEDDruleareexamplesofstaticpriorityrules.ArulethatissomewhatrelatedtotheEDDruleistheso-calledMinimumSlackfirst(MS)rulewhichselects,whenthemachinebecomesavailableattimet,thejobwiththeleast
5.2SingleMachineandParallelMachineModels85slack;theslackofjobjattimetisdefinedasmax(dj−pj−t,0).ThisruledoesnotoperateinexactlythesamewayastheEDDrule,butwillresultinschedulesthataresomewhatsimilartoEDDschedules.However,theMSruleisanexampleofadynamicpriorityrule,inwhichthepriorityofeachjobisafunctionoftime.Otherobjectives,suchasthetotaltardinessTjandthetotalweightedtardinesswjTj,aremuchhardertooptimizethanthetotalweightedcom-pletiontimeorthemaximumlateness.AheuristicforthetotalweightedtardinessobjectiveisdescribedinAppendixC.ThisheuristicisreferredtoastheApparentTardinessCostfirst(ATC)rule.Ifthemachineisfreedattimet,theATCruleselectsamongtheremainingjobsthejobwiththehighestvalueofIj(t)=wjpjexp−max(dj−pj−t,0)K¯p,whereKisaso-calledscalingparameterand¯pistheaverageoftheprocessingtimesofthejobsthatremaintobescheduled.TheATCpriorityruleisactuallyaweightedmixtureoftheWSPTandMSpriorityrulesmentionedabove.ByadjustingthescalingparameterKtherulecanbemadetooperateeithermoreliketheWSPTruleormoreliketheMSrule.Whenthejobsinasinglemachineproblemhavedifferentreleasedatesrj,thentheproblemstendtobecomesignificantlymorecomplicated.Onefamousproblemthathasreceivedasignificantamountofattentionisthenonpreemp-tivesinglemachineschedulingprobleminwhichthejobshavedifferentreleasedatesandthemaximumlatenessLmaxhastobeminimized.ThisproblemisNP-Hard,whichimpliesthat,unfortunately,noefficient(polynomialtime)al-gorithmexistsforthisproblem.Theproblemcanbesolvedeitherbybranch-and-boundorbydynamicprogramming(abranch-and-boundmethodforthisproblemisdescribedinAppendixB).Itturnsoutthataprocedureforsolv-ingthissinglemachineproblemisanimportantstepwithinthewell-knownshiftingbottleneckheuristicforgeneraljobshops.WhentherearemmachinesinparallelthemakespanCmaxisscheduledependent.Themakespanobjective,whichplaysanimportantrolewhentheloadsofthevariousmachineshavetobebalanced,givesrisetoanotherpriorityrule,namelytheLongestProcessingTimefirst(LPT)rule.Accordingtothisrule,wheneveroneofthemachinesisfreed,thelongestjobamongthosewaitingforprocessingisselectedtogonext.Theintuitionbehindthisruleisclear:Onewouldliketoprocessthesmallerjobstowardstheendoftheschedulesincethatmakesiteasiertobalancethemachines.Ifthesmallerjobsgolast,thenthelongerjobshavetobescheduledmoreinthebeginning.TheLPTrule,unfortunately,doesnotguaranteeanoptimalsolution(itcanbeshownthatthisruleguaranteesasolutionthatiswithin33%oftheoptimalsolution,seeExercise5.3).TheSPTandtheWSPTrulesareimportantintheparallelmachineset-tingaswell.Ifallnjobsareavailableatt=0,thenthenonpreemptiveSPTruleminimizesthetotalcompletiontimeCj;thenonpreemptiveSPTrule
865MachineSchedulingandJobShopSchedulingremainsoptimalevenwhenpreemptionsareallowed.Unfortunately,minimiz-ingthetotalweightedcompletiontimewjCjinaparallelmachinesettingwhenalljobsareavailableatt=0isNP-Hard,sotheWSPTruledoesnotminimizethewjCjinthiscase.However,theWSPTrulestillcanbeusedasaheuristicanditisguaranteedtogenerateasolutionthatiswithin22%ofoptimality.Themoregeneralparallelmachineproblemwiththetotalweightedtar-dinessobjective(wjTj)isevenharder.TheATCruledescribedaboveisapplicabletotheparallelmachinesettingaswell;however,thequalityofthesolutionsmayattimesleavesomethingtobedesired.5.3JobShopsandMathematicalProgrammingConsiderajobshopwithnjobsandmmachines.Eachjobhastobeprocessedbyanumberofmachinesinagivenorderandthereisnorecirculation.Theprocessingofjobjonmachineiisreferredtoasoperation(i,j)anditsdurationispij.TheobjectiveistominimizethemakespanCmax.Theproblemofminimizingthemakespaninajobshopwithoutrecircula-tioncanberepresentedbyaso-calleddisjunctivegraph.ConsideradirectedgraphGwithasetofnodesNandtwosetsofarcsAandB.ThenodesNcorrespondtoalloftheoperations(i,j)thatmustbeperformedonthenjobs.Theso-calledconjunctive(solid)arcsArepresenttheroutesofthejobs.Ifarc(i,j)→(h,j)ispartofA,thenjobjhastobeprocessedonmachineibeforeitisprocessedonmachineh,i.e.,operation(i,j)precedesoperation(h,j).Twooperationsthatbelongtotwodifferentjobsandwhichhavetobedisjunctive(broken)arcsgoinginoppositedirections.ThedisjunctivearcsBformmcliquesofdoublearcs,onecliqueforeachmachine.(Acliqueisatermingraphtheorythatreferstoagraphinwhichanytwonodesareconnectedtooneanother;inthiscaseeachconnectionwithinacliqueisapairofdisjunctivearcs.)Alloperations(nodes)inthesamecliquehavetobedoneonthesamemachine.Allarcsemanatingfromanode,conjunctiveaswellasdisjunctive,haveaslengththeprocessingtimeoftheoperationthatisrepresentedbythatnode.InadditionthereisasourceUandasinkV,whicharedummynodes.ThesourcenodeUhasnconjunctivearcsemanatingtothefirstoperationsofthenjobsandthesinknodeVhasnconjunctivearcscominginfromallthelastoperations.Thearcsemanatingfromthesourcehavelengthzero,seeFigure5.1.WedenotethisgraphbyG=(N,A,B).Afeasibleschedulecorrespondstoaselectionofonedisjunctivearcfromeachpairsuchthattheresultingdirectedgraphisacyclic.Thisimpliesthateachselectionofarcsfromwithinacliquemustbeacyclic.Suchaselectiondeterminesthesequenceinwhichtheoperationsaretobeperformedonthatmachine.Thataselectionfromacliquehastobeacycliccanbearguedasprocessedonthesamemachineareconnectedtooneanotherbytwoso-called
5.3JobShopsandMathematicalProgramming871, 1p23Sink02, 13, 12, 21, 24, 21, 32, 34, 3U3, 2VSourcep42p43p4300Fig.5.1.Directedgraphforjobshopwiththemakespanastheobjectivefollows:Iftherewereacyclewithinaclique,afeasiblesequenceoftheopera-tionsonthecorrespondingmachinewouldnothavebeenpossible.Itmaynotbeimmediatelyobviouswhythereshouldnotbeanycycleformedbycon-junctivearcsanddisjunctivearcsfromdifferentcliques.Suchacyclewouldalsocorrespondtoasituationthatisinfeasible.Forexample,let(h,j)and(i,j)denotetwoconsecutiveoperationsthatbelongtojobjandlet(i,k)and(h,k)denotetwoconsecutiveoperationsthatbelongtojobk.Ifunderagivenscheduleoperation(i,j)precedesoperation(i,k)onmachineiandoperation(h,k)precedesoperation(h,j)onmachineh,thenthegraphcontainsacyclewithfourarcs,twoconjunctivearcsandtwodisjunctivearcsfromdifferentcliques.Suchascheduleisphysicallyimpossible.Summarizing,ifDdenotesthesubsetoftheselecteddisjunctivearcsandthegraphG(D)isdefinedbythesetofconjunctivearcsandthesubsetD,thenDcorrespondstoafeasiblescheduleifandonlyifG(D)containsnodirectedcycles.ThemakespanofafeasiblescheduleisdeterminedbythelongestpathinG(D)fromthesourceUtothesinkV.Thislongestpathconsistsofasetofoperationsofwhichthefirststartsattime0andthelastfinishesatthetimeofthemakespan.Eachoperationonthispathisimmediatelyfollowedeitherbythenextoperationonthesamemachineorbythenextoperationofthesamejobonanothermachine.Theproblemofminimizingthemakespanisreducedtofindingaselectionofdisjunctivearcsthatminimizesthelengthofthelongestpath(i.e.,thecriticalpath).Thereareseveralmathematicalprogrammingformulationsforthejobshopwithoutrecirculation,includinganumberofintegerprogrammingformula-tions,seeSection4.6andExercise5.4.However,theformulationmostoftenusedistheso-calleddisjunctiveprogrammingformulation,seeAppendixA.Thisdisjunctiveprogrammingformulationiscloselyrelatedtothedisjunctivegraphrepresentationofthejobshop.Topresentthedisjunctiveprogrammingformulation,letthevariableyijdenotethestartingtimeofoperation(i,j).RecallthatsetNdenotesthesetofalloperations(i,j),andsetAthesetofallroutingconstraints(i,j)→(h,j)
885MachineSchedulingandJobShopSchedulingwhichrequirejobjtobeprocessedonmachineibeforeitisprocessedonmachineh.Thefollowingmathematicalprogramminimizesthemakespan.minimizeCmaxsubjecttoyhj−yij≥pijforall(i,j)→(h,j)∈ACmax−yij≥pijforall(i,j)∈Nyij−yik≥pikoryik−yij≥pijforall(i,k)and(i,j),i=1,…,myij≥0forall(i,j)∈NInthisformulation,thefirstsetofconstraintsensurethatoperation(h,j)cannotstartbeforeoperation(i,j)iscompleted.Thethirdsetofconstraintsarecalledthedisjunctiveconstraints;theyensurethatsomeorderingexistsamongoperationsofdifferentjobsthathavetobeprocessedonthesamemachine.Becauseoftheseconstraintsthisformulationisreferredtoasthedisjunctiveprogrammingformulation.Example5.3.1(DisjunctiveProgrammingFormulation).Considerthefollowingexamplewithfourmachinesandthreejobs.Theroute,i.e.,themachinesequence,aswellastheprocessingtimesarepresentedinthetablebelow.jobsmachinesequenceprocessingtimes11,2,3p11=10,p21=8,p31=422,1,4,3p22=8,p12=3,p42=5,p32=631,2,4p13=4,p23=7,p43=3TheobjectiveconsistsofthesinglevariableCmax.Thefirstsetofconstraintsconsistsofsevenconstraints:twoforjob1,threeforjob2andtwoforjob3.Forexample,oneoftheseisy21−y11≥10(=p11).Thesecondsetconsistsoftenconstraints,oneforeachoperation.AnexampleisCmax−y11≥10(=p11).Thesetofdisjunctiveconstraintscontainseightconstraints:threeeachformachines1and2andoneeachformachines3and4(therearethreeoperationstobeperformedonmachines1and2andtwooperationsonmachines3and4).Anexampleofadisjunctiveconstraintisy11−y12≥3(=p12)ory12−y11≥10(=p11).Thelastsetincludestennonnegativityconstraints,oneforeachstartingtime.
5.4JobShopsandtheShiftingBottleneckHeuristic89Thataschedulingproblemcanbeformulatedasadisjunctiveprogramdoesnotimplythatthereisastandardsolutionprocedureavailablethatwillalwaysworksatisfactorily.Minimizingthemakespaninajobshopisaveryhardproblemandsolutionproceduresareeitherbasedonenumerationoronheuristics.Tofindoptimalsolutionsbranch-and-boundmethodshavetobeused.AppendixBprovidesanexampleofabranch-and-boundprocedureappliedtoajobshop.5.4JobShopsandtheShiftingBottleneckHeuristicSinceithasturnedouttobeveryhardtosolvejobshopproblemswithlargenumbersofjobstooptimality,manyheuristicprocedureshavebeendesigned.OneofthemostsuccessfulproceduresforminimizingthemakespaninajobshopistheShiftingBottleneckheuristic.InthefollowingoverviewoftheShiftingBottleneckheuristicMdenotesthesetofallmmachines.InthedescriptionofaniterationoftheheuristicitisassumedthatinpreviousiterationsaselectionofdisjunctivearcshasbeenfixedforasubsetM0ofmachines.SoforeachoneofthemachinesinM0asequenceofoperationshasalreadybeendetermined.AniterationresultsinaselectionofamachinefromM−M0forinclu-sioninsetM0.Thesequenceinwhichtheoperationsonthismachinearetobeprocessedisalsogeneratedinthisiteration.Todeterminewhichma-chineshouldbeincludednextinM0,anattemptismadetodeterminewhichunscheduledmachinecausesinonesenseoranothertheseverestdisruption.Todeterminethis,theoriginaldirectedgraphismodifiedbydeletingalldis-junctivearcsofthemachinesstilltobescheduled(i.e.,themachinesinsetM−M0)andkeepingonlytherelevantdisjunctivearcsofthemachinesinsetM0(onefromeverypair).CallthisgraphG.Deletingalldisjunctivearcsofaspecificmachineimpliesthatalloperationsonthismachine,whichoriginallyweresupposedtobedoneonthismachineoneafteranother,nowmaybedoneinparallel(asifthemachinehasinfinitecapacity,orequivalently,eachoneoftheseoperationshasthemachineforitself).ThegraphGhasoneormorecriticalpathsthatdeterminethecorrespondingmakespan.CallthismakespanCmax(M0).Supposethatoperation(i,j),i∈{M−M0},hastobeprocessedinatimewindowofwhichthereleasedateandduedatearedeterminedbythecritical(longest)pathsinG,i.e.,thereleasedateisequaltothelongestpathinGfromthesourcetonode(i,j)andtheduedateisequaltoCmax(M0),minusthelongestpathfromnode(i,j)tothesink,pluspij.ConsidereachofthemachinesinM−M0asaseparatenonpreemptivesinglemachineproblemwithreleasedatesandduedatesandwiththemaximumlatenesstobemin-imized.Asstatedintheprevioussection,thisproblemisNP-hard;however,procedureshavebeendevelopedthatperformreasonablywell.Theminimum
905MachineSchedulingandJobShopSchedulingLmaxofthesinglemachineproblemcorrespondingtomachineiisdenotedbyLmax(i)andisameasureofthecriticalityofmachinei.Aftersolvingallthesesinglemachineproblems,themachinewiththelargestmaximumlatenessLmax(i)isselected.Amongtheremainingmachines,thismachineisinasensethemostcriticalorthe“bottleneck”andthereforetheonetobeincludednextinM0.Assumethisismachineh,callitsmaximumlatenessLmax(h)andscheduleitaccordingtotheoptimalsolutionobtainedforthesinglemachineproblemassociatedwiththismachine.IfthedisjunctivearcsthatspecifythesequenceofoperationsonmachinehareinsertedingraphG,thenthemakespanofthecurrentpartialscheduleincreasesbyatleastLmax(h),thatis,Cmax(M0∪h)≥Cmax(M0)+Lmax(h).Beforestartingthenextiterationanddeterminingthenextmachinetobescheduled,anadditionalstephastobedonewithinthecurrentiteration.InthisadditionalstepallthemachinesintheoriginalsetM0areresequencedonebyoneinordertoseeifthemakespancanbereduced.Thatis,ama-chine,saymachinel,istakenoutofsetM0andagraphGisconstructedbymodifyinggraphGthroughtheinclusionofthedisjunctivearcsthatspecifythesequenceofoperationsonmachinehandtheexclusionofthedisjunctivearcsassociatedwithmachinel.Machinelisresequencedbysolvingthecorre-spondingsinglemachinemaximumlatenessproblemwiththereleaseandduedatesdeterminedbythecriticalpathsingraphG.ResequencingeachofthemachinesintheoriginalsetM0completestheiteration.InthenextiterationtheentireprocedureisrepeatedandanothermachineisaddedtothecurrentsetM0∪h.Theshiftingbottleneckheuristiccanbesummarizedasfollows.Algorithm5.4.1(ShiftingBottleneckHeuristic).Step1.(Initialconditions)SetM0=∅.GraphGisthegraphwithalltheconjunctivearcsandnodisjunctivearcs.SetCmax(M0)equaltothelongestpathingraphG.Step2.(Analysisofmachinesstilltobescheduled)DoforeachmachineiinsetM−M0thefollowing:formulateasinglemachineproblemwithalloperationssubjecttoreleasedatesandduedates(thereleasedateofoperation(i,j)isdeterminedbythelongestpathingraphGfromthesourcetonode(i,j);theduedateofoperation(i,j)canbecomputedbyconsideringthelongestpathingraphGfromnode(i,j)tothesinkandsubtractingpij).MinimizetheLmaxineachoneofthesesinglemachinesubproblems.LetLmax(i)denotetheminimumLmaxinthesubproblemcorrespondingtomachinei.
5.4JobShopsandtheShiftingBottleneckHeuristic91Step3.(Bottleneckselectionandsequencing)LetLmax(h)=maxi∈{M−M0}(Lmax(i))SequencemachinehaccordingtothesequencegeneratedforitinStep2.InsertallthecorrespondingdisjunctivearcsingraphG.InsertmachinehinM0.Step4.(Resequencingofallmachinesscheduledearlier)Doforeachmachinel∈{M0−h}thefollowing:DeletethecorrespondingdisjunctivearcsfromG;formulateasinglema-chinesubproblemformachinelwithreleasedatesandduedatesoftheoperationsdeterminedbylongestpathcalculationsinG.FindthesequencethatminimizesLmax(l)andinsertthecorrespondingdisjunctivearcsingraphG.Step5.(Stoppingcriterion)IfM0=MthenSTOP,otherwisegotoStep2.Thestructureoftheshiftingbottleneckheuristicshowstherelationshipbetweenthebottleneckconceptandthemorecombinatorialconceptssuchascritical(longest)pathandmaximumlateness.Acriticalpathindicatesthelocationandthetimingofabottleneck.Themaximumlatenessgivesanindicationoftheamountbywhichthemakespanincreasesifamachineisaddedtothesetofmachinesalreadyscheduled.Thefollowingexampleillustratestheuseoftheshiftingbottleneckheuristic.Example5.4.2(ApplicationofShiftingBottleneckHeuristic).Con-sidertheinstancewithfourmachinesandthreejobsdescribedinExamples5.3.1.Therouting,i.e.,themachinesequences,andtheprocessingtimesaregiveninthefollowingtable:jobsmachinesequenceprocessingtimes11,2,3p11=10,p21=8,p31=422,1,4,3p22=8,p12=3,p42=5,p32=631,2,4p13=4,p23=7,p43=3Iteration1:Initially,setM0isemptyandgraphGcontainsonlyconjunc-tivearcsandnodisjunctivearcs.ThecriticalpathandthemakespanCmax(∅)canbedeterminedeasily:thismakespanisequaltothemaximumtotalpro-cessingtimerequiredforanyjob.Themaximumof22isachievedinthiscasebybothjob1andjob2.Todeterminewhichmachinetoschedulefirst,eachmachineisconsideredasanonpreemptivesinglemachinemaximumlateness
925MachineSchedulingandJobShopScheduling1, 102, 13, 12, 21, 24, 21, 32, 34, 3U3, 2V0010881033357446Fig.5.2.Iteration1ofshiftingbottleneckheuristic(Example5.4.2)problemwiththereleasedatesandduedatesdeterminedbythelongestpathsinG(assumingamakespanof22).Thedataforthenonpreemptivesinglemachinemaximumlatenessproblemcorrespondingtomachine1isdeterminedasfollows.jobs123p1j1034r1j080d1j101112Theoptimalsequenceturnsouttobe1,2,3withLmax(1)=5.Thedataforthesubproblemassociatedwithmachine2are:jobs123p2j887r2j1004d2j18819Theoptimalsequenceforthisproblemis2,3,1withLmax(2)=5.Similarly,itcanbeshownthatLmax(3)=4andLmax(4)=0.Fromthisitfollowsthateithermachine1ormachine2maybeconsideredabottleneck.Breakingthetiearbitrarily,machine1isselectedtobeincludedinM0.ThegraphGisobtainedbyfixingthedisjunctivearcscorrespondingtothesequenceofthejobsonmachine1(seeFigure5.2).ItisclearthatCmax({1})=Cmax(∅)+Lmax(1)=22+5=27.
5.4JobShopsandtheShiftingBottleneckHeuristic93Iteration2:GiventhatthemakespancorrespondingtoGis27,thecriticalpathsinthegraphcanbedetermined.Thethreeremainingmachineshavetobeanalyzedseparatelyasnonpreemptivesinglemachineproblems.Thedatafortheproblemconcerningmachine2are:jobs123p2j887r2j10017d2j231024Theoptimalscheduleis2,1,3andtheresultingLmax(2)=1.Thedatafortheproblemcorrespondingtomachine3are:jobs12p3j46r3j1818d3j2727BothsequencesareoptimalandLmax(3)=1.Machine4canbeanalyzedinthesamewayandtheresultingLmax(4)=0.Again,thereisatieandmachine2isselectedtobeincludedinM0.SoM0={1,2}andCmax({1,2})=Cmax({1})+Lmax(2)=27+1=28.Thedisjunctivearcscorrespondingtothejobsequenceonmachine2areaddedtoGandgraphGisobtained.Atthispoint,stillasapartofiteration2,anattemptmaybemadetodecreaseCmax({1,2})byresequencingmachine1.Itcanbecheckedthatresequencingmachine1doesnotgiveanyimprovement.Iteration3:ThecriticalpathinGcanbedeterminedandmachines3and4remaintobeanalyzed.Thesetwoproblemsturnouttobeverysimplewithbothhavingazeromaximumlateness.Neitherofthemachinesconstitutesabottleneckinanyway.Thefinalscheduleisdeterminedbythefollowingmachinesequences:thejobsequence1,2,3onmachine1;thejobsequence2,1,3onmachine2;thejobsequence2,1onmachine3andthejobsequence2,3onmachine4.Themakespanis28.Thesinglemachinemaximumlatenessproblemthathastobesolvedre-peatedlywithineachiterationoftheshiftingbottleneckheuristicmayactuallybeslightlydifferentandmorecomplicatedthanthesinglemachinemaximumlatenessproblemdescribedinSection5.2andinExampleB.4.1.InthesinglemachinesubproblemthathastobesolvedinStep2oftheshiftingbottleneckheuristic,theoperationsonthegivenmachinemaybesubjecttoaspecialtypeofprecedenceconstraints.Itmaybethatanoperationonthemachinetobescheduledmustbeprocessedaftercertainotheroperationshavebeen
945MachineSchedulingandJobShopSchedulingcompletedonthatmachine;theseprecedenceconstraintsmustbesatisfiedbecauseofthesequencesofoperationsonthemachinesthatalreadyhavebeenscheduledinpreviousiterations.Moreover,itmayevenbethecasethatinbetweentheprocessingoftwooperationssubjecttotheseprecedencecon-straintsacertainminimumamountoftime(i.e.,adelay)hastoelapse.Thelengthsofthedelaysarealsodeterminedbythesequencesoftheoperationsonthemachinesalreadyscheduledinpreviousiterations.Theseprecedenceconstraintsarethereforereferredtoasdelayedprecedenceconstraints.Itiseasytoconstructexamplesthatshowtheneedfordelayedprecedenceconstraintsinthesinglemachinesubproblem.Withouttheseconstraintstheshiftingbottleneckheuristicmayendupinasituationwithacycleinthedisjunctivegraphandthecorrespondingschedulebeinginfeasible.ExtensivenumericalresearchhasshownthattheShiftingBottleneckheuristicisextremelyeffective.Whenappliedtoafamoustestproblemwith10machinesand10jobsthathadremainedunsolvedformorethan20years,theheuristicfoundaverygoodsolutionveryfast.Thissolutionactuallyturnedouttobeoptimalafterabranch-and-boundprocedurefoundthesameresultandverifieditsoptimality.Thebranch-and-boundapproach,incontrasttotheheuristic,neededmanyhoursofCPUtime.Thedisadvantageoftheheuristicisthatthereisneveraguaranteethatthesolutiongeneratedisoptimal.TheShiftingBottleneckheuristiccanbeextendedtomodelsthataremoregeneralthanthebasicjobshopconsideredabove.Itcanbeappliedtomoregeneralmachineenvironmentsandprocessingcharacteristicsaswellastootherobjectivefunctions.Thedisjunctivegraphformulationforthejobshopproblemwithoutrecir-culationalsoextendstothejobshopproblemwithrecirculation.Thesetofdis-junctivearcsforamachinethatissubjecttorecirculationisnownotaclique.Iftwooperationsofthesamejobhavetobeperformedonthesamemachineaprecedencerelationshipisgiven.Thesetwooperationsarenotconnectedbyapairofdisjunctivearcs,sincetheyarealreadyconnectedbyconjunctivearcs.TheShiftingBottleneckheuristicdescribedcanstillbeimplemented.Thejobsinthesinglemachinesubproblemarenowsubjecttoprecedenceconstraints,whichareimposedbythefactthatdifferentoperationsofthesamejobmayrequireprocessingonthesamemachine.Themostgeneralmachineenvironmentthatcanbedealtwithistheflex-iblejobshop.Eachworkcenterconsistsofasetofmachinesinparallel.Thedisjunctivegraphrepresentationcanbeextendedtothismachinesettingasfollows:again,allpairsofoperationsthathavetobeperformedatawork-centerarelinkedbypairsofdisjunctivearcs.Iftwooperationsarescheduledforprocessingonthesamemachine,thentheappropriatedisjunctivearcisselected.Iftwooperationsareassignedtodifferentmachinesinthework-center,thenbothdisjunctivearcsaredeletedfromthegraph,sincethetwooperationsdonotaffecteachother’sprocessing.ThesubproblemthatthenhastobesolvedinStep2oftheShiftingBottleneckHeuristic(wichwasintheoriginalversionoftheShiftingBottleneckheuristicasinglemachinemax-
5.5JobShopsandConstraintProgramming95imumlatenessproblemwithjobshavingdifferentreleasedates)nowbecomesaparallelmachinemaximumlatenessproblemwithjobsthathavedifferentreleasedates.Therearevariationsoftheshiftingbottleneckheuristicthatcanbeap-pliedtojobshopproblemswiththetotalweightedtardinessobjective.Thedisjunctivegraphformulationforthejobshopwiththetotalweightedtar-dinessobjectiveisslightlydifferentfromthedisjunctivegraphformulationforthemakespanobjective.Also,theobjectivefunctionofthesubproblemisamorecomplicatedobjectivethantheLmaxobjectiveinthesubproblemdescribedabove.Thetechniquedescribedabovecanalsobeadaptedtoflexiblejobshopswithmultipleobjectives.Wehavealreadyseenthatthemakespanobjectiveintheoriginaljobshopproblemleadstoamaximumlatenessobjectiveinitssinglemachinesubproblems.Thetotalweightedtardinessintheoriginaljobshopproblemleadstoamorecomplicatedduedaterelatedobjectivefunctioninthesinglemachinesubproblems.Multipleobjectivesintheoriginalproblemleadtomultipleobjectivesinthesinglemachine(orparallelmachines)sub-problems;theweightsofthevariousobjectivesintheoriginalproblemhave,ofcourse,animpactontheweightsoftheobjectivesinthesubproblems.TheheuristicapproachesdescribedabovecanbelinkedtolocalsearchproceduresasdescribedinAppendixC.Forexample,theshiftingbottle-neckheuristiccanbeusedfirsttoobtainaschedulewithareasonablygoodmakespanandthissolutionisthenfedintoalocalsearchprocedurethatmayyieldanevenbettersolution.5.5JobShopsandConstraintProgrammingConstraintprogrammingisatechniquethatoriginatedintheArtificialIn-telligence(AI)community.Inrecentyears,ithasoftenbeenimplementedinconjunctionwithOperationsResearch(OR)techniquesinordertoimproveitsoveralleffectiveness(seeAppendixD).Constraintprogramming,initsoriginaldesign,onlytriestofindagoodsolutionthatisfeasibleandsatisfiesallthegivenconstraints.Thesolutionsfoundmay,therefore,notnecessarilybeoptimal.Theconstraintsmayin-cludereleasedatesandduedatesforthejobs.Itispossibletoembedsuchatechniqueinaframeworkthatisdesignedtoactuallyminimizeanyduedaterelatedobjectivefunction.Constraintprogrammingcanbeappliedtothebasicjobshopwiththemakespanobjectiveasfollows.SupposethatinajobshopaschedulehastobefoundwithamakespanCmaxthatislessthanorequaltoagivendeadline¯d.Aconstraintsatisfactionalgorithmhastoproduceforeachmachineasequenceoftheoperationsinsuchawaythattheoverallschedulehasamakespanthatislessthanorequalto¯d.
965MachineSchedulingandJobShopSchedulingBeforetheactualprocedurestarts,aninitializationstephastobedone.Foreachoperationacomputationisdonetodetermineitsearliestpossiblestartingtimeandlatestpossiblecompletiontimeonthemachineinques-tion.Afterallthetimewindowshavebeencomputed,thetimewindowsofalltheoperationsoneachmachinearecomparedwithoneanother.Whenthetimewindowsoftwooperationsonanygivenmachinedonotoverlap,aprecedencerelationshipbetweenthetwooperationscanbeimposed;inanyfeasiblescheduletheoperationwiththeearliertimewindowmustprecedetheoperationwiththelatertimewindow.Actually,aprecedencerelationshipmayevenbeinferredwhenthetimewindowsdooverlap.LetSij(Sij)denotetheearliest(latest)possiblestartingtimeofoperation(i,j)andCij(Cij)theearliest(latest)possiblecompletiontimeofoperation(i,j)underthecurrentsetofprecedenceconstraints.Notethattheearliestpossiblestartingtimeofoperation(i,j),i.e.,Sij,mayberegardedasalocalreleasedateoftheopera-tionandmaybedenotedbyrij,whereasthelatestpossiblecompletiontime,i.e.,Cij,maybeconsideredalocalduedatedenotedbydij.Definetheslackbetweentheprocessingofoperations(i,j)and(i,k)onmachineiasσ(i,j)→(i,k)=Sik−Cij=Cik−Sij−pij−pik=dik−rij−pij−pik.Ifσ(i,j)→(i,k)<0then,underthecurrentsetofprecedenceconstraints,nofeasiblescheduleexistsinwhichoperation(i,j)precedesoperation(i,k)onmachinei.Soaprecedencerelationshipcanbeimposedthatrequiresoperation(i,k)toappearbeforeoperation(i,j).Intheinitializationstepoftheprocedureallpairsoftimewindowsarecomparedwithoneanotherandallimpliedprecedencerelationshipsareinsertedinthedisjunctivegraph.Becauseoftheseadditionalprecedenceconstraintsthetimewindowsofeachoneoftheoperationscanbeadjusted(narrowed),i.e.,thisinvolvesarecomputationofthereleasedateandtheduedateofeachoperation.Constraintsatisfactiontechniquesoftenrelyonconstraintpropagation.Aconstraintsatisfactiontechniquetypicallyattempts,ineachstep,toinsertnewprecedenceconstraints(disjunctivearcs)thatareimpliedbytheprecedenceconstraintsinsertedbefore.Withthenewprecedenceconstraintsinplacethetechniquerecomputesthetimewindowsofalloperations.Foreachpairofoperationsthathavetobeprocessedonthesamemachineithastobeverifiedwhichoneofthefollowingfourcasesholds:Case1:Ifσ(i,j)→(i,k)≥0andσ(i,k)→(i,j)<0,thentheprecedenceconstraint(i,j)→(i,k)hastobeimposed. 5.5JobShopsandConstraintProgramming97Case2:Ifσ(i,k)→(i,j)≥0andσ(i,j)→(i,k)<0,thentheprecedenceconstraint(i,k)→(i,j)hastobeimposed.Case3:Ifσ(i,j)→(i,k)<0andσ(i,k)→(i,j)<0,thenthereisnoschedulethatsatisfiestheprecedenceconstraintsalreadyinplace.Case4:Ifσ(i,j)→(i,k)≥0andσ(i,k)→(i,j)≥0,theneitherorderingbetweenthetwooperationsisstillpossible.Inonestepofthealgorithmthatisdescribedlateroninthissection,apairofoperationshastobeconsideredthatsatisfytheconditionsofCase4,i.e.,eitherorderingbetweentheoperationsisstillpossible.ItmaybethecasethatinthisstepofthealgorithmmanypairsofoperationsstillsatisfyCase4.IfthereismorethanonepairofoperationsthatsatisfytheconditionsofCase4,thenaselectionheuristichastobeapplied.Theselectionofapairisbasedonthesequencingflexibilitythispairstillprovides.Thepairwiththelowestflexibilityisselected.Thereasoningbehindthisapproachisstraightforward:ifapairwithlowflexibilityisnotscheduledearlyonintheprocess,thenlateronintheprocessthatpairmaynotbeschedulableatall.Soitmakessensetogiveprioritytothosepairswithalowflexibilityandpostponepairswithahighflexibility.Clearly,theflexibilitydependsontheamountsofslackinthetwoorderings.Onesimpleestimateofthesequencingflexibilityofapairofoperations,φ((i,j)(i,k)),istheminimumofthetwoslacks,i.e.,φ((i,j)(i,k))=minσ(i,j)→(i,k),σ(i,k)→(i,j).However,relyingonthisminimummayleadtoproblems.Forexample,sup-poseonepairofoperationshasslackvalues3and100,whereasanotherpairhasslackvalues4and4.Inthiscase,theremaybeonlylimitedpossibilitiesforschedulingthesecondpairandpostponingadecisionwithregardtothesecondpairmaywelleliminatethem.Afeasibleorderingofthefirstpairmaynotreallybeinjeopardy.Insteadofusingφ((i,j)(i,k))thefollowingmeasureofsequencingflexibilityhasproventobemoreeffective:φ((i,j)(i,k))=min(σ(i,j)→(i,k),σ(i,k)→(i,j))×max(σ(i,j)→(i,k),σ(i,k)→(i,j)).Soifthemaxislarge,thentheflexibilityofapairofoperationsincreasesandtheurgencytoorderthepairgoesdown.Afterthepairofoperationswiththeleastflexibilityφ((i,j)(i,k))hasbeenselected,theprecedenceconstraintthatretainsthemostflexibilityisimposed,i.e.,ifσ(i,j)→(i,k)≥σ(i,k)→(i,j) 985MachineSchedulingandJobShopSchedulingoperation(i,j)mustprecedeoperation(i,k).InoneofthestepsofthealgorithmitalsocanhappenthatapairofoperationssatisfiesCase3.Whenthisisthecasethepartialschedulethatisunderconstructioncannotbecompletedandthealgorithmhastobacktrack.Backtrackingcantakeanyoneofseveralforms.Backtrackingmayimplythateither(i)oneormoreoftheorderingdecisionsmadeinearlieriterationshastobeannulled,or(ii)theredoesnotexistafeasiblesolutionfortheprobleminthewayitwasformulatedandoneormoreoftheoriginalconstraintsintheproblemhavetoberelaxed.Theconstraintsatisfactionprocedurecanbesummarizedasfollows.Algorithm5.5.1(ConstraintSatisfactionProcedure).Step1.Computeforeachunorderedpairofoperationstheslacksσ(i,j)→(i,k)andσ(i,k)→(i,j).Step2.Checkdominanceconditionsandclassifyremainingorderingdecisions.IfanyorderingdecisionisofCase3,thenBACKTRACK.IfanyorderingdecisioniseitherofCase1orCase2gotoStep3;otherwisegotoStep4.Step3.InsertnewprecedenceconstraintandgotoStep1.Step4.IfnoorderingdecisionisofCase4,thensolutionisfound.STOP.OtherwisegotoStep5.Step5.Computeφ((i,j)(i,k))foreachpairofoperationsnotyetordered.Selectthepairwiththeminimumφ((i,j)(i,k)).Ifσ(i,j)→(i,k)≥σ(i,k)→(i,j),thenoperation(i,k)mustfollow(i,j);otherwiseoperation(i,j)mustfollowoperation(i,k).GotoStep3.Inordertoapplytheconstraintsatisfactionproceduretoajobshopprob-lemwiththemakespanobjective,ithastobeembeddedinthefollowingframework.First,anupperboundduandalowerbounddlhavetobefoundforthemakespan. 5.5JobShopsandConstraintProgramming991, 102, 13, 12, 21, 24, 21, 32, 34, 3U3, 2V0010883357446Fig.5.3.DisjunctivegraphwithoutdisjunctivearcsAlgorithm5.5.2(FrameworkforConstraintProgramming).Step1.Setd=(dl+du)/2.ApplyAlgorithm5.5.1.Step2.IfCmaxd,setdl=d.Step3.Ifdu−dl>1returntoStep1.OtherwiseSTOP.Thefollowingexampleillustratestheuseofthecontraintsatisfactiontech-nique.Example5.5.3(ApplicationofConstraintProgrammingtoaJobShop).ConsidertheinstanceofthejobshopproblemdescribedinExample5.3.1.jobsmachinesequenceprocessingtimes11,2,3p11=10,p21=8,p31=422,1,4,3p22=8,p12=3,p42=5,p32=631,2,4p13=4,p23=7,p43=3Consideraduedated=32whenalljobshavetobecompleted.Consideragainthedisjunctivegraphbutdisregardalldisjunctivearcs,seeFigure5.3.Bydoingalllongestpathcomputations,thelocalreleasedatesandlocalduedatesforalloperationscanbeestablished.
1005MachineSchedulingandJobShopSchedulingoperationsrijdij(1,1)020(2,1)1028(3,1)1832(2,2)018(1,2)821(4,2)1126(3,2)1632(1,3)022(2,3)429(4,3)1132Giventhesetimewindowsforalltheoperations,ithastobeveri-fiedwhethertheseconstraintsalreadyimplyanyadditionalprecedencecon-straints.Consider,forexample,thepairofoperations(2,2)and(2,3)whichbothhavetogoonmachine2.Computingtheslackyieldsσ(2,3)→(2,2)=d22−r23−p22−p23=18−4−8−7=−1,whichimpliesthattheordering(2,3)→(2,2)isnotfeasible.Sothedisjunc-tivearc(2,2)→(2,3)hastobeinserted.Inthesameway,itcanbeshownthatthedisjunctivearcs(2,2)→(2,1)and(1,1)→(1,2)havetobeinsertedaswell.Giventheseadditionalprecedenceconstraintsthereleaseandduedatesofalloperationshavetobeupdated.Theupdatedreleaseandduedatesarepresentedinthetablebelow.operationsrijdij(1,1)018(2,1)1028(3,1)1832(2,2)018(1,2)1021(4,2)1326(3,2)1832(1,3)022(2,3)829(4,3)1532Theseupdatedreleaseandduedatesdonotimplyanyadditionalprece-denceconstraints.GoingthroughStep5ofthealgorithmrequiresthecom-putationofthefactorφ((i,j)(i,k))foreveryunorderedpairofoperationsoneachmachine.
5.5JobShopsandConstraintProgramming101pairφ((i,j)(i,k))(1,1)(1,3)√4×8=5.65(1,2)(1,3)√5×14=8.36(2,1)(2,3)√4×5=4.47(3,1)(3,2)√4×4=4.00(4,2)(4,3)√3×11=5.74Thepairwiththeleastflexibilityis(3,1)(3,2).Sincetheslacksaresuchthatσ(3,2)→(3,1)=σ(3,1)→(3,2)=4,eitherprecedenceconstraintcanbeinserted.Supposetheprecedencecon-straint(3,2)→(3,1)isinserted.Thisprecedenceconstraintcausessignif-icantchangesinthetimewindowsduringwhichtheoperationshavetobeprocessed.operationsrijdij(1,1)014(2,1)1028(3,1)2432(2,2)014(1,2)1017(4,2)1322(3,2)1828(1,3)022(2,3)829(4,3)1532However,thisnewsetoftimewindowsimposesanadditionalprecedenceconstraint,namely(4,2)→(4,3).Thisnewprecedenceconstraintcausesthefollowingchangesinthereleasedatesandduedatesoftheoperations.operationsrijdij(1,1)014(2,1)1028(3,1)2432(2,2)014(1,2)1017(4,2)1322(3,2)1828(1,3)022(2,3)829(4,3)1832
1025MachineSchedulingandJobShopScheduling0102030t1, 11, 31, 22, 22, 12, 33, 23, 14, 24, 3Fig.5.4.FinalscheduleinExample5.5.3.Theseupdatedreleaseandduedatesdonotimplyadditionalprecedenceconstraints.GoingthroughStep5ofthealgorithmrequiresthecomputationofthefactorφ((i,j)(i,k))foreveryunorderedpairofoperationsoneachmachine.pairφ((i,j)(i,k))(1,1)(1,3)√0×8=0.00(1,2)(1,3)√5×10=7.07(2,1)(2,3)√4×5=4.47Thepairwiththeleastflexibilityis(1,1)(1,3)andtheprecedencecon-straint(1,1)→(1,3)hastobeinserted.Insertingthislastprecedenceconstraintenforcesonemoreconstraint,namely(2,1)→(2,3).Nowonlyoneunorderedpairofoperationsremains,namelypair(1,3)(1,2).Thesetwooperationscanbeorderedineitherwaywithoutviolatinganyduedates.Afeasibleorderingis(1,3)→(1,2).Theresultingschedulewithamakespanof32isdepictedinFigure5.4.Thisschedulemeetstheduedateoriginallysetbutisnotoptimal.Whenthepair(3,1)(3,2)hadtobeordered,itcouldhavebeenorderedineitherdirectionbecausethetwoslackvalueswereequal.Supposeatthatpointtheoppositeorderingwasselected,i.e.,(3,1)→(3,2).Restartingtheprocessatthatpointyieldsthefollowingreleaseandduedates.
5.5JobShopsandConstraintProgramming103operationsrijdij(1,1)014(2,1)1022(3,1)1826(2,2)018(1,2)1021(4,2)1326(3,2)1832(1,3)022(2,3)829(4,3)1532Thesereleaseandduedatesenforceaprecedenceconstraintonthepairofoperations(2,1)(2,3)andtheconstraintis(2,1)→(2,3).Thisadditionalconstrainthasthefollowingeffectonthereleaseandduedates:operationsrijdij(1,1)014(2,1)1022(3,1)1826(2,2)018(1,2)1021(4,2)1326(3,2)2232(1,3)022(2,3)1829(4,3)2532Thesenewreleaseandduedateshaveaneffectonthepair(4,2)(4,3)andthearc(4,2)→(4,3)hastobeincluded.Thisadditionalarcdoesnotcauseanyadditionalchangesinthereleaseandduedates.Atthispointonlytwopairsofoperationsremainunordered,namelythepair(1,1)(1,3)andthepair(1,2)(1,3).pairφ((i,j)(i,k))(1,1)(1,3)√0×8=0.00(1,2)(1,3)√5×14=8.36Sothepair(1,1)(1,3)ismorecriticalandhastobeordered(1,1)→(1,3).Itturnsoutthatthelastpairtobeordered,(1,2)(1,3),canbeorderedeitherway.Theresultingscheduleturnsouttobeoptimalandhasamakespanof28,seeFigure5.5.
1045MachineSchedulingandJobShopSchedulingMachine 1Machine 2Machine 3Machine 40102030t1,1 1,31,22,22,12,33,13,24,24,3Fig.5.5.AlternativescheduleonExample5.5.3Asstatedbefore,constraintsatisfactionisnotonlysuitableformakespanminimization;itcanalsobeappliedtoproblemswithduedaterelatedobjec-tivesandalsowheneachjobhasitsownreleasedate.5.6LEKIN:AGenericJobShopSchedulingSystemSincethejobshopisaverycommonmachineenvironment,manyschedulingsystemshavebeendevelopedforthisenvironment.OneexampleofsuchasystemistheLEKINsystem.Originally,thissystemwasdesignedasatoolforteachingandresearch.Eventhoughtheoriginalversionwasdesignedwithacademicgoalsinmind,severaloffshootsarebeingusedinrealworldimple-mentations.TheacademicversionofthesystemisavailableontheCD-ROMthatisattachedtothisbook.Thesystemcontainsanumberofschedulingalgorithmsandheuristicsandisdesignedtoallowtheusertolinkandtesthisorherownheuristicsandcomparetheirperformanceswiththeheuristicsandalgorithmsthatareembeddedinthesystem.TheLEKINsystemcanaccomodatevariousmachineenvironments,namely:(i)singlemachine(ii)parallelmachines(iii)flowshop(iv)flexibleflowshop(v)jobshop(vi)flexiblejobshopFurthermore,itiscapableofdealingwithsequencedependentsetuptimesinalltheenvironmentslistedabove.
5.6LEKIN:AGenericJobShopSchedulingSystem105Inthemainmenutheusercanselectamachineenvironment.Aftertheuserhasselectedanenvironment,hehastoenterallthenecessarymachinedataandjobdatamanually.However,inthemainmenutheuseralsohastheoptionofopeninganexistingdatafile.Anexistingfilecontainsdatawithregardtooneofthemachineenvironmentsandaspecificsetofjobs.Theusercanopenanexistingfile,makechangesinthefileandworkwiththemodifiedfile.Attheendofthesessiontheusercansavethemodifiedfileunderanewname.Iftheuserwantstoenteradatasetthatiscompletelynew,hefirstmustselectamachineenvironment,andthenadialogboxappearswherehehastoenterthemostbasicinformation,i.e.,thenumberofworkcentersandthenumberofjobstobescheduled.Aftertheuserhasdonethis,aseconddialogboxappearsandhehastoenterthemoredetailedworkcenterinformation,i.e.,thenumberofmachinesattheworkcenter,theiravailability,andthedetailsneededtodeterminethesetuptimesoneachmachine(iftherearesetuptimes).Inthethirddialogboxtheuserhastoenterthedetailedinformationwithregardtothejobs,i.e.,releasedates,duedates,prioritiesorweights,routing,andtheprocessingtimesofthevariousoperations.Ifthejobsrequiresequencedependentsetuptimes,thenthemachinesettingsthatarerequiredfortheprocessinghavetobeentered.IntheLEKINsystemtherequiredmachinesettingisequivalenttoasetuptimeparameter;asequencedependentsetuptimeisthenafunctionoftheparameterofthejobjustcompletedandtheparameterofthejobabouttobestarted.Afterallthedatahavebeenenteredfourwindowsappearsimultaneously,namely,(i)themachineparkwindow,(ii)thejobpoolwindow,(iii)thesequencewindow,and(iv)theGanttchartwindow,(seeFigure5.6).Themachineparkwindowdisplaysalltheinformationregardingthework-centersandthemachines.Thisinformationisorganizedintheformatofatree.Thiswindowfirstshowsalistofalltheworkcenters.Iftheuserclicksonaworkcenter,theindividualmachinesofthatworkcenterappear.Thejobpoolwindowcontainsthestartingtime,completiontime,andmoreinformationwithregardtoeachjob.Theinformationwithregardtothejobsisalsoorganizedintheformofatree.First,thejobsarelisted.Iftheuserclicksonaspecificjob,thenimmediatelyalistofthevariousoperationsthatbelongtothatjobappear.Thesequencewindowcontainsthelistsofjobsintheorderinwhichtheyareprocessedoneachoneofthevariousmachines.Thepresentationherealsohasatreestructure.First,allthemachinesarelisted.Iftheuserclicksonamachine,thenalltheoperationsthatareprocessedonthatmachineappearinthesequenceinwhichtheyareprocessed.ThiswindowisequivalenttothedispatchlistinterfacedescribedinChapter13.Atthebottomofthis
1065MachineSchedulingandJobShopSchedulingFig.5.6.FourwindowsoftheLEKINsystemsequencewindowthereisasummaryofthevariousperformancemeasuresofthecurrentschedule.TheGanttchartwindowcontainsaconventionalGanttchart.ThisGanttchartwindowenablestheusertodoanumberofthings.Forexample,theusercanclickonanoperationandawindowpopsupdisplayingthedetailedinformationwithregardtothecorrespondingjob(seeFigure5.7).TheGanttchartwindowalsohasabuttonthatactivatesawindowwheretheusercanseethecurrentvaluesofalltheobjectives.Thewindowsdescribedabovecanbedisplayedsimultaneouslyonthescreeninanumberofways,e.g.,inaquadrantstyle,tiledhorizontally,ortiledvertically(seeFigure5.6).Besidesthesefourwindowstherearetwootherwindows,whichwillbedescribedinmoredetaillater.Thesetwowin-dowsarethelogbookwindowandtheobjectivechartwindow.Theusercanprintoutthewindowsseparatelyoralltogetherbyselectingtheprintoptionintheappropriatewindow.Thedatasetofaparticularschedulingproblemcanbemodifiedinanumberofways.First,informationwithregardtotheworkcenterscanbemodifiedinthemachineparkwindow.Whentheuserdouble-clicksontheworkcentertherelevantinformationappears.Machineinformationcanbeac-
5.6LEKIN:AGenericJobShopSchedulingSystem107Fig.5.7.Ganttchartwindowcessedinasimilarmanner.Jobscanbeadded,modified,ordeletedinthejobpoolwindow.Adoubleclickonajobdisplaysalltherelevantinforma-tion.Aftertheuserhasenteredthedataset,alltheinformationisdisplayedinthemachineparkwindowandjobpoolwindow.However,thesequencewindowandtheGanttchartwindowremainempty.Iftheuserinthebegin-ninghadopenedanexistingfile,thenthesequencewindowandtheGanttchartwindowmaydisplayinformationpertainingtoasequencethathadbeengeneratedduringanearliersession.Theusercanselectaschedulefromanywindow.TypicallytheuserwoulddosoeitherfromthesequencewindoworfromtheGanttchartwindowbyclickingonscheduleandselectingaheuristicoralgorithmfromthedrop-downmenu.AscheduleisthengeneratedanddisplayedinboththesequencewindowandtheGanttchartwindow.TheschedulegeneratedanddisplayedintheGanttchartisaso-calledsemi-activeschedule.Asemi-activescheduleischaracterizedbythefactthatthestarttimeofanyoperationofanygivenjobonanygivenmachineisequaltoeitherthecompletiontimeoftheprecedingoperationofthesamejobonadifferentmachineorthecompletiontimeofanoperationofadifferentjobonthesamemachine.
1085MachineSchedulingandJobShopSchedulingThesystemcontainsanumberofalgorithmsforseveralofthemachineenvironmentsandobjectivefunctions.Thesealgorithmsinclude(i)dispatchingrules,(ii)heuristicsoftheshiftingbottlenecktype,(iii)localsearchtechniques,and(iv)aheuristicfortheflexibleflowshopwiththetotalweightedtardinessasobjective(SB-LS).ThedispatchingrulesincludeEDDandWSPT.Thewaythesedispatchingrulesareappliedinasinglemachineenvironmentandinaparallelmachineenvironmentisstandard.However,theycanalsobeappliedinthemorecom-plicatedmachineenvironmentssuchastheflexibleflowshopandtheflexiblejobshop.Theyarethenappliedasfollows:eachtimeamachineisfreedthesystemcheckswhichjobshouldgonextonthatmachine.Thesystemthenusesthefollowingdataforitspriorityrules:theduedateofacandidatejobisthentheduedateatwhichthejobhastoleavethesystem.TheprocessingtimethatispluggedintheWSPTruleisthesumoftheprocessingtimesofalltheremainingoperationsofthatjob.Thesystemalsohasageneralpurposeroutineoftheshiftingbottlenecktypethatcanbeappliedtoeachoneofthemachineenvironmentsandeveryobjectivefunction.Sincethisroutineisquitegenericanddesignedformanydifferentmachineenvironmentsandobjectivefunctions,itcannotcompeteagainstashiftingbottleneckheuristicthatistailor-madeforaspecificmachineenvironmentandaspecificobjectivefunction.Thesystemalsocontainsaneighbourhoodsearchroutinethatisapplicabletotheflowshopandjobshop(butnottotheflexibleflowshoporflexiblejobshop)witheitherthemakespanorthetotalweightedtardinessasobjective.Iftheuserselectstheshiftingbottleneckorthelocalsearchoption,thenhemustselectalsotheobjectivehewantstominimize.Whentheuserselectsthelocalsearchoptionandtheobjective,awindowappearsinwhichtheuserhastoenterthenumberofsecondshewantsthelocalsearchtorun.Thesystemalsohasaspecializedroutinefortheflexibleflowshopwiththetotalweightedtardinessasobjective;thisroutineisacombinationofashiftingbottleneckroutineandalocalsearch(SB-LS).Thisroutinetendstoyieldschedulesthatareofreasonablyhighquality.Iftheuserwantstoconstructaschedulemanually,hecandosoinoneoftwoways.First,hecanmodifyanexistingscheduleintheGanttchartwindowasmuchashewantsbyclicking,dragginganddroppingoperations.Aftersuchmodificationstheresultingscheduleis,again,semi-active.However,theusercanalsoconstructthiswayaschedulethatisnotsemi-active.TodothishehastoactivatetheGanttchart,holddowntheshiftbuttonandmovetheoperationtothedesiredposition.Whentheuserreleasestheshiftbutton,theoperationremainsfixed.Asecondwayofcreatingaschedulemanuallyisthefollowing.Afterclick-ingonschedule,theusermustselect“manualentry”.Theuserthenhasto
5.6LEKIN:AGenericJobShopSchedulingSystem109Fig.5.8.Logbookandcomparisonofdifferentschedulesenterforeachmachineajobsequence.Thejobsinasequenceareseparatedfromoneanotherbya“;”.Whenevertheusergeneratesascheduleforaparticulardataset,thesched-uleisstoredinalogbook.Thesystemautomaticallyassignsanappropriatenametoeveryschedulegenerated.Thesystemcanstoreandretrieveanumberofdifferentschedules,seeFigure5.8.Iftheuserwantstocomparethedifferentscheduleshehastoclickon“logbook”.Theusercanchangethenameofeachscheduleandgiveeachscheduleadifferentnameforfuturereference.Theschedulesstoredinthelogbookcanbecomparedwithoneanotherbyclickingonthe“performancemeasures”button.Theusermaythenselectoneormoreobjectives.Iftheuserselectsasingleobjective,abarchartappearsthatcomparestheschedulesstoredinthelogbookwithrespecttotheobjectiveselected.Iftheuserwantstocomparethescheduleswithregardtotwoobjectives,an(x,y)coordinatesystemappearsandeachscheduleisrepresentedbyadot.Iftheuserselectsthreeormoreobjectives,thenamulti-dimensionalgraphappearsthatdisplaystheperformancemeasuresoftheschedulesstoredinthelogbook.Someusersmaywanttoincorporatetheconceptofsetuptimes.Iftherearesetuptimes,thentherelevantdatahavetobeenteredtogetherwithallother
1105MachineSchedulingandJobShopSchedulingFig.5.9.Setuptimematrixjobandmachinedataattheverybeginningofasession.(However,ifatthebeginningofasessionanexistingfileisopened,thensuchafilemayalreadycontainsetupdata.)Thesetuptimesarebasedonthefollowingstructure.Eachoperationhasasingleparameterorattribute,whichisrepresentedbyaletter,e.g.,A,B,andsoon.Thisparameterrepresentsthemachinesettingrequiredforprocessingthatoperation.Whentheuserentersthedataforeachmachine,hehastofillinasetuptimematrixforthatmachine.Thesetuptimematrixforamachinespecifiesthetimethatittakestochangethatmachinefromonesettingtoanother,i.e.,fromBtoE,seeFigure5.9.Thesetuptimematricesofallthemachinesatanygivenworkcenterhavetobethesame(themachinesataworkcenterareassumedtobeidentical).Thissetuptimestructuredoesnotallowtheusertoimplementarbitrarysetuptimematrices.Amoreadvancedusercanlinkhisownalgorithmstothesystem.ThisfeatureallowsthedeveloperofanewalgorithmtotesthisalgorithmusingtheinteractiveGanttchartfeaturesofthesystem.Theprocessofmakingsuchanexternalprogramrecognizablebythesystemconsistsoftwosteps,namely,thepreparationofthecode(programming,compilinganddebugging),andthelinkingofthecodetothesystem.
Exercises111Thelinkingofanexternalprogramisdonebyclickingon“Tools”andselecting“Options”.AwindowappearswithabuttonforaNewFolderandabuttonforaNewItem.ClickingonNewfoldercreatesanewsubmenu.ClickingonaNewItemcreatesaplaceholderforanewalgorithm.AftertheuserhasclickedonaNewItem,hehastoenterallthedatawithrespecttotheNewItem.Under“Executable”hehastoenterthefullpathtotheexecutablefileoftheprogram.ThedevelopmentofthecodecanbedoneinanyenvironmentunderWin3.2.AppendixFprovidesexamplesoftheformatofafilethatcontainstheworkcenterinformationandtheformatofafilethatcontainstheinformationpertainingtothejob.Afteranewalgorithmhasbeenadded,itisincludedasoneoftheheuristicsinthescheduleoptionmenu.5.7DiscussionAnenormousamountofresearchonmachineschedulingandjobshopschedul-inghasresultedinmanybooks.Thischapterjustgivesaflavorofthetypesofresultsthathaveappearedintheliterature.Itfocusesmainlyonthetypeofresearchthathasproventobeusefulinpractice,i.e.,decompositiontech-niquessuchastheshiftingbottleneckheuristicandconstraintprogrammingtechniques.Asignificantamountofmoretheoreticalresearchhasappearedinthelit-eraturefocusingonexactoptimizationtechniquesbasedonintegerprogram-minganddisjunctiveprogrammingformulations.SomeoftheseformulationsarepresentedinAppendixA.Thetechniquesdevelopedincludebranch-and-boundaswellasbranch-and-cut.Anexampleofabranch-and-boundappli-cationispresentedinAppendixB.Someofthemoreappliedtechniqueshavenotbeenincludedinthischaptereither.Afairamountofresearchhasbeendoneonlocalsearchtechniquesap-pliedtomachineschedulingandjobshopscheduling.Thesetechniquesincludesimulatedannealing,tabu-searchaswellasgeneticalgorithms.ExamplesofsuchapplicationsaregiveninAppendixC.AppendixDcontainsaconstraintprogrammingformulationintheOPLlanguageofajobshopproblemwiththemakespanobjective.Exercises5.1.WhatdoestheATCrulereduceto(a)whenKgoesto∞,and(b)whenKisveryclosetozero?5.2.Consideraninstancewithtwomachinesinparallelandthetotalweightedtardinessasobjectivetobeminimized.Thereare5jobs.
1125MachineSchedulingandJobShopSchedulingjobs12345pj13913108dj618101113wj24254(a)ApplytheATCheuristiconthisinstancewiththelook-aheadparam-eterK=1.(b)ApplytheATCheuristiconthisinstancewiththelook-aheadparam-eterK=5.5.3.Consider6machinesinparalleland13jobs.Theprocessingtimesofthe13jobsaretabulatedbelow.jobs12345678910111213pj66677889910101111(a)ComputethemakespanunderLPT.(b)Findtheoptimalschedule.5.4.ExplainwhytheproblemdiscussedinSection5.3isaspecialcaseoftheproblemdescribedinSection4.6.(Hint:Consideranarbitraryjobshopschedulingproblemanddescribeitasaworkforceconstrainedprojectschedul-ingproblem.Consideramachineinthejobshopschedulingproblemasaworkforcepoolintheprojectschedulingproblemandtheavailablenumberinthepoolbeingequalto1.)5.5.Considerthefollowinginstanceofthejobshopproblemwithnorecircu-lationandthemakespanasobjective.jobsmachinesequenceprocessingtimes11,2,3p11=9,p21=8,p31=421,2,4p12=5,p22=6,p42=333,1,2p33=10,p13=4,p23=9Giveanintegerprogrammingformulationofthisinstance(Hint:ConsidertheintegerprogrammingformulationoftheprojectschedulingproblemwithworkforceconstraintsdescribedinSection4.6andExercise5.4.)5.6.Considerthefollowingheuristicforthejobshopproblemwithnorecir-culationandthemakespanobjective.Eachtimeamachineisfreed,selectthejob(amongthoseimmediatelyavailableforprocessingonthemachine)withthelongesttotalremainingprocessing(includingitsprocessingonthema-chinefreed).Ifatanypointintimemorethanonemachineisfreed,considerfirstthemachinewiththelargestremainingworkload.ApplythisheuristictotheinstanceinExample5.3.1.
CommentsandReferences1135.7.ApplytheheuristicdescribedinExercise5.6tothefollowinginstancewiththemakespanobjective.jobsmachinesequenceprocessingtimes11,2,3,4p11=9,p21=8,p31=4,p41=421,2,4,3p12=5,p22=6,p42=3,p32=633,1,2,4p33=10,p13=4,p23=9,p43=25.8.ConsidertheinstanceinExercise5.7.(a)ApplytheShiftingBottleneckheuristictothisinstance(doingthecomputationbyhand).(b)CompareyourresultwiththeresultoftheshiftingbottleneckroutineintheLEKINsystem.5.9.Considerthefollowingtwomachinejobshopwith10jobs.Alljobshavetobeprocessedfirstonmachine1andthenonmachine2.(Thisimpliesthatthetwomachinejobshopisactuallyatwomachineflowshop).jobs1234567891011p1j364342755612p2j45523366472(a)ApplytheheuristicdescribedinExercise5.6tothistwomachinejobshop.(b)Constructnowascheduleasfollows.Thejobshavetogothroughthesecondmachineinthesamesequenceastheygothroughthefirstmachine.Ajobwhoseprocessingtimeonmachine1isshorterthanitsprocessingtimeonmachine2mustprecedeeachjobwhoseprocessingtimeonmachine1islongerthanitsprocessingtimeonmachine2.Thejobswithashorterprocessingtimeonmachine1aresequencedinincreasingorderoftheirprocessingtimesonmachine1.Thejobswithashorterprocessingtimeonmachine2followindecreasingorderoftheirprocessingtimesonmachine2.(ThisruleisusuallyreferredtoasJohnson’sRule.)(c)ComparethescheduleobtainedwithJohnson’sruletothescheduleobtainedunder(a).5.10.ApplytheshiftingbottleneckheuristictothetwomachineflowshopinstanceinExercise5.9.ComparetheresultingschedulewiththeschedulesobtainedinExercise5.9.CommentsandReferencesManyofthebasicpriorityrulesoriginatedinthefiftiesandearlysixties.Forexam-ple,Smith(1956)introducedtheWSPTrule,Jackson(1955)introducedtheEDDrule,andHu(1961)theCPrule.Conway(1965a,1965b)wasoneofthefirsttomake
1145MachineSchedulingandJobShopSchedulingacomparativestudyofpriorityrules.AfairlycompletelistofthemostcommonpriorityordispatchingrulesisgivenbyPanwalkarandIskander(1977)andade-taileddescriptionofcompositedispatchingrulesbyBhaskaranandPinedo(1992).SpecialexamplesofcompositedispatchingrulesaretheCOVERTruledevelopedbyCarroll(1965)andtheATCruledevelopedbyVepsalainenandMorton(1987).OwandMorton(1989)describearuleforschedulingproblemswithearlinessandtardinesspenalties.TheATCSruleisduetoLee,BhaskaranandPinedo(1997).Thetotalweightedtardinesshasbeenthefocusofanumberofstudiesinthemorebasicmachineenvironments,e.g.,thesinglemachine.Therearemanyre-searcherswhohavedealtwiththesinglemachinetotalweightedtardinessproblem;see,forexample,Fisher(1976),PottsandVanWassenhove(1982,1985,1987),andRachamadugu(1987).Abdul-Razaq,Potts,andVanWassenhove(1990)presentasurveyofalgorithmsforthesinglemachinetotalweightedtardinessproblem.Vep-salainenandMorton(1987)developedpriorityrulebasedheuristicsforjobshopswiththetotalweightedtardinessobjective.Jobshopschedulinghasreceivedanenormousamountofattentioninthere-searchliteratureaswellasinbooks.Thespecialcaseofthejobshopwhereallthejobshavethesameroute,i.e.,theflowshop,alsohasreceivedconsiderableatten-tion.Forresultsontheflowshop,see,forexample,Johnson(1954),Palmer(1965),Campbell,DudekandSmith(1970),Szwarc(1971,1973,1978),Gupta(1972),Baker(1975),Smith,PanwalkarandDudek(1975,1976),Dannenbring(1977),Muth(1979),PapadimitriouandKannelakis(1980),Ow(1985),Pinedo(1985),WidmerandHertz(1989),andTaillard(1990).AnalgorithmfortheminimizationofthemakespaninatwomachinejobshopwithoutrecirculationisduetoJackson(1956)andthedisjunctiveprogrammingformulationdescribedinSection5.3isduetoRoyandSussmann(1964).Branch-and-boundtechniqueshaveoftenbeenappliedtominimizethemakespaninjobshops;see,forexample,Lomnicki(1965),BrownandLomnicki(1966),McMa-honandFlorian(1975),BarkerandMcMahon(1985),CarlierandPinson(1989),ApplegateandCook(1991),andHoitomt,LuhandPattipati(1993).Foranoverviewofbranch-and-boundtechniques,seePinson(1995).SingerandPinedo(1998)de-velopedabranch-and-boundapproachforthejobshopwiththetotalweightedtardinessobjective.ThefamousshiftingbottleneckheuristicisduetoAdams,BalasandZawack(1988).Theiralgorithmmakesuseofasinglemachineschedulingalgorithmdevel-opedbyCarlier(1982).EarlierworkonthisparticularsinglemachineschedulingproblemwasdonebyMcMahonandFlorian(1975).NowickiandZdrzalka(1986),Dauz`ere-P´er`esandLasserre(1993,1994)andBalas,LenstraandVazacopoulos(1995)alldevelopedmoresophisticatedversionsoftheCarlieralgorithm.Themono-graphbyOvacikandUzsoy(1997)presentsanexcellenttreatiseoftheapplicationofdecompositionmethodsandshiftingbottlenecktechniquestolargescalejobshopswiththemakespanandthemaximumlatenessobjectives.Thismonographisbasedonanumberofpapersbythetwoauthors;see,forexample,Uzsoy(1993)fortheap-plicationofdecompositionmethodstoflexiblejobshops.PinedoandSinger(1998)developedashiftingbottleneckapproachforthejobshopproblemwiththetotalweightedtardinessobjectiveandYang,Kreipl,andPinedo(2000)developedasim-ilarapproachfortheflexibleflowshopwiththetotalweightedtardinessobjective.Afairamountofresearchhasfocusedonconstraintprogrammingapproachesforthejobshopschedulingproblemwiththemakespanobjective;Section5.5is
CommentsandReferences115basedontheworkbyChengandSmith(1997).Severalotherresearchershavealsoconsideredthisproblem;see,forexample,NuijtenandAarts(1996),andBaptiste,LePape,andNuijten(2001).TheLEKINsystemisduetoAsadathorn(1997)andFeldmanandPinedo(1998).Thegeneral-purposeroutineoftheshiftingbottlenecktypethatisembeddedinthesystemisalsoduetoAsadathorn(1997).Thelocalsearchroutinesthatareappli-cabletotheflowshopandjobshopareduetoKreipl(1998).ThemorespecializedSB-LSroutinefortheflexibleflowshopisduetoYang,Kreipl,andPinedo(2000).Inadditiontotheproceduresdiscussedinthischapterflowshopandjobshopproblemshavebeentackledwithlocalsearchprocedures;see,forexample,Matsuo,Suh,andSullivan(1988),Dell’AmicoandTrubian(1991),DellaCroce,TadeiandVolta(1992),Storer,WuandVaccari(1992),NowickiandSmutnicki(1996),andKreipl(1998).Forabroaderviewofthejobshopschedulingproblem,seeWeinandChevelier(1992).Foraninterestingspecialcaseofthejobshop,i.e.,aflowshopwithreentry,seeGraves,Meal,StefekandZeghmi(1983).Forresultsonageneralizationoftheflowshop,i.e.,theflexibleflowshop,seeHodgsonandMcDonald(1981a,1981b,1981c).Ofcourse,manyapplicationsofjobshopschedulingproblemsinindustryhavebeendiscussedintheliterature.Forjobshopschedulingproblemsandsolutionsinthemicro-electronicsindustry,see,forexample,Wein(1988),Lee,UzsoyandMartin-Vega(1992)andUzsoy,LeeandMartin-Vega(1992).
Chapter6SchedulingofFlexibleAssemblySystems6.1Introduction……………………………1176.2SequencingofUnpacedAssemblySystems……1186.3SequencingofPacedAssemblySystems……..1246.4SchedulingofFlexibleFlowSystemswithBypass1296.5MixedModelAssemblySequencingatToyota..1346.6Discussion……………………………..1376.1IntroductionFlexibleassemblysystemsdifferinanumberofwaysfromthejobshopsconsideredinthepreviouschapter.Inajobshop,eachjobhasitsownidentityandmaybedifferentfromallotherjobs.Inaflexibleassemblysystem,therearetypicallyalimitednumberofdifferentproducttypesandthesystemhastoproduceagivenquantityofeachproducttype.Sotwounitsofthesameproducttypeareidentical.Themovementsofjobsinaflexibleassemblysystemareoftencontrolledbyamaterialhandlingsystem,whichimposesconstraintsonthestartingtimesofthejobsatthevariousmachinesorworkstations.Thestartingtimeofajobatamachineisafunctionofitscompletiontimeonthepreviousmachineonitsroute.Amaterialhandlingsystemusuallyalsolimitsthenumberofjobswaitinginbuffersbetweenmachines.Inthischapterweanalyzethreedifferentmodelsforflexibleassemblysystems.Themachineenvironmentsinthethreemodelsaresimilartothemachineenvironmentsofaflowshoporaflexibleflowshop.However,themodelstendtobemorecomplicatedthanthemodelsconsideredinChapter5.Thisismainlybecauseoftheadditionalconstraintsthatareimposedbythematerialhandlingsystems.© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_6,117
1186SchedulingofFlexibleAssemblySystemsThefirstmodelrepresentsaflowlinewithanumberofmachinesorwork-stationsinseries.Thelineisunpaced,i.e.,amachinecanspendasmuchtimeasneededonanyjob.Therearefinitebuffersbetweensuccessivema-chineswhichmaycauseblockingandstarving.Anumberofdifferentproducttypeshavetobeproducedingivenquantitiesandthegoalistomaximizethethroughput.Thistypeofenvironmentoccurs,forexample,intheassemblyofcopiers.Differenttypesofcopiersareoftenassembledonthesameline.Thedifferentmodelsareusuallyfromthesamefamilyandmayhavemanycom-moncharacteristics.However,theydifferwithregardtotheoptions.Sometypeshaveanautomaticdocumentfeederwhileothershavenot;somehavemoreelaborateopticsthanothers.Thefactthatdifferentcopiershavediffer-entoptionsimpliesthattheprocessingtimesatcertainstationsmayvary.Thesecondmodelisapacedassemblysystemwithaconveyorsystemthatmovesataconstantspeed.Theunitsthathavetobeassembledaremovedfromoneworkstationtothenextatafixedspeed.Eachworkstationhasitsowncapacityandconstraints.Again,anumberofdifferentproducttypeshavetobeassembled.Thegoalistosequencethejobssothatnoworkstationisoverloadedandsetupcostsareminimized.Pacedassemblysystemsareverycommonintheautomotiveindustry,wheredifferentmodelshavetobeassembledonthesameline.Thecarsmayhavedifferentcolorsanddifferentoptionpackages.Thesequencingofthecarshastotakesetupcostsaswellasworkloadbalancingintoaccount.Thethirdmodelisaflexibleflowsystemwithlimitedbuffersandbypass.Incontrasttothefirsttwomodels,thereareanumberofmachinesinparallelateachworkcenter.Ajobmaybeprocessedonanyoneoftheparallelmachinesoritmaybypassaworkcenteraltogether.Theobjectiveistomaximizethethroughput.Thistypeofsystemisused,forexample,inthemanufacturingofPrintedCircuitBoards(PCBs).PCBsareproducedinbatcheswitheachbatchrequiringitsownsetofoperations.6.2SequencingofUnpacedAssemblySystemsConsideranumberofmachinesinserieswithalimitedbufferbetweensuc-cessivemachines.Thematerialhandlingsystemthatmovesajobfromonemachinetothenextisunpaced.Sowheneveramachinefinishesprocessingajobitcanreleasethejobtothebufferofthenextmachineprovidedthatbufferisnotfull.Ifthatbufferisfull,thenthemachinecannotreleasethejobandisblocked.Thematerialhandlingsystemdoesnotpermitbypassing,i.e.,eachmachineservesthejobsaccordingtotheFirstInFirstOutdiscipline.Thistypeofserialprocessingiscommonintheassemblyofbulkyitems,suchastelevisionsetsorcopiers;theirsizemakesitdifficulttokeepmanyunitswaitinginfrontofamachine.Thismachineenvironmentwithlimitedbuffersisinamathematicalsenseequivalenttoasystemofmachinesinserieswithnobuffersinbetweenma-
6.2SequencingofUnpacedAssemblySystems119chines.Thishappenstobethecasebecauseabufferspacebetweentwoma-chinesmayberegardedasa”machine”wheretheprocessingtimesofalljobshappentobezero.Soanynumberofmachineswithlimitedbuffersbetweenthemcanbetransformedintoanequivalentsystemthatconsistsofa(larger)numberofmachineswithzerobuffersinbetweenthem.Transformingamodelwithbuffersintoanequivalentmodelwithoutbuffersisadvantageous,sinceamodelwithoutbuffersiseasiertodescribeandtoformulatemathematicallythanamodelwithbuffers.Inthissectionwefocus,therefore,onmodelswith-outbuffers;ofcourse,wemayhavemachineswhereallprocessingtimesarezero(playingtheroleofbuffers).Schedulesusedinflowlineswithblockingareoftenperiodicorcyclic.Suchschedulesaregeneratedasfollows.First,agivensetofjobsarescheduledinacertainorder.Thissetcontainsjobsofalltheproducttypesandtheremaybemorethanonejobofthesameproducttype.Thissetisfollowedbyasecondsetthatisidenticaltothefirstoneandscheduledinthesameway.Thisprocessisrepeatedoverandoveragain;theresultingscheduleisacyclicschedule.Cyclicschedulesaregenerallynotpracticalwhentherearesignificantse-quencedependentsetuptimesbetweendifferentproducttypes.Itisthenmoreefficienttohavelongrunsofthesameproducttype.However,ifsetuptimesarenegligible,thencyclicscheduleshaveimportantadvantages.Forexam-ple,ifthedemandforeachproducttyperemainsconstantovertime,thenacyclicscheduleresultsininventoryholdingcoststhataresignificantlylessthanthecostsincurredwithschedulesthathavelongrunsofeachproducttype.Cyclicschedulesalsohaveanadvantageinthattheyareeasytokeeptrackofandimposeacertaindiscipline.However,itisnotnecessarilytruethatacyclicschedulehasthemaximumthroughput;often,anacyclicsched-ulehasthemaximumthroughput.Nevertheless,inpractice,ascheduleroftenadoptsacyclicschedulefromwhichheallowsminordeviations,dependentuponcurrentdemand.Supposethereareldifferentproducttypes.LetNkdenotethenumberofjobsofproducttypekintheoverallproductiontarget.Theproductiontargetmaybeforaperiodofsixmonthsoroneyearandthenumbersmaybelarge.IfzisthegreatestcommondivisoroftheintegersN1,…,Nl,thenthevectorN∗=N1z,…,Nlzrepresentsthesmallestsethavingthesameproportionsofthedifferentprod-ucttypesasthelongrangeproductiontarget.ThissetisusuallyreferredtoastheMinimumPartSet(MPS).GiventhevectorN∗,theelementsinanMPSmayberegarded,withoutlossofgenerality,asnjobs,wheren=1zlk=1Nk.
1206SchedulingofFlexibleAssemblySystemsLetpijdenotetheprocessingtimeofjobj,j=1,…,n,onmachinei.CyclicschedulesarespecifiedbythesequenceofthenjobsintheMPS.ThefactthatvariousjobswithinanMPSmaycorrespondtothesameproducttypeandhaveidenticalprocessingrequirementsdoesnotaffecttheapproachdescribedbelow.MaximizingsystemthroughputisbasicallyequivalenttominimizingthecycletimeofanMPSinasteadystate.InthiscasetheMPScycletimecanbedefinedasthetimebetweenthefirstjobsoftwoconsecutiveMPSsenteringthesystem.ThefollowingexampleillustratestheMPScycletimeconcept.Example6.2.1(MPSCycleTime).Consideranassemblysystemwithfourmachinesandnobuffersbetweenthemachines.Therearethreedifferentproducttypesthathavetobeproducedinequalamounts,i.e.,N∗=(1,1,1).TheprocessingtimesofthethreejobsintheMPSare:Jobs123p1j010p2j000p3j101p4j110Thesecond“machine”(thatis,thesecondrowofprocessingtimes),withzeroprocessingtimesforallthreejobs,functionsasabufferbetweenthefirstandthirdmachines.TheGanttchartsforthisexampleunderthetwodifferentsequencesareshowninFigure6.1.Underbothsequencesasteadystateisreachedduringthesecondcycle.Undersequence1,2,3theMPScycletimeisthree,whileundersequence1,3,2theMPScycletimeistwo.FortheminimizationoftheMPScycletimetheso-calledProfileFitting(PF)heuristiccanbeused.Thisheuristicworksasfollows:onejobisselectedtogofirst.TheselectionofthefirstjobintheMPSsequencemaybedonearbitrarilyoraccordingtosomescheme.Forexample,onecanchoosethejobwiththelargesttotalamountofprocessingastheonetogofirst.Thisfirstjobgeneratesaprofile.Forthetimebeing,weassumethatthejobdoesnotencounteranyblockingandproceedssmoothlyfromonemachinetothenext(inasteadystatethefirstjobinanMPSmaybeblockedbythelastjobfromthepreviousMPS).Theprofileisdeterminedbythedeparturetimeofthisfirstjob,sayjobj1,frommachinei.IfXi,j1denotesthedeparture(orexit)timeofjobj1frommachinei,thenXi,j1=ih=1ph,j1
6.2SequencingofUnpacedAssemblySystems121Cycle timeCycle timeCycle timeCycle time1231231312312123123111232311Fig.6.1.GanttchartsforExample6.2.1Inordertodeterminewhichisthemostappropriatejobtogosecond,everyoneoftheremainingjobsintheMPSistriedout.Foreachcandidatejob,theamountsoftimethemachinesareidleandtheamountsoftimethejobisblockedatamachinearecomputed.Thedeparturetimesofacandidatejobforthesecondposition,sayjobk,canbecomputedrecursivelyasfollows:X1,j2=max(X1,j1+p1k,X2,j1)Xi,j2=max(Xi−1,j2+pik,Xi+1,j1),i=2,…,m−1Xm,j2=Xm−1,j2+pmkThenonproductivetimeonmachinei,eitherfrombeingidleorfrombeingblocked,ifcandidatekisputintothesecondposition,isXi,j2−Xi,j1−pik.Thesumoftheseidleandblockedtimesoverallmmachinesiscomputedforcandidatek,i.e.,
1226SchedulingofFlexibleAssemblySystemsmi=1Xi,j2−Xi,j1−pik.ThisprocedureisrepeatedforallremainingjobsintheMPS.Thecandidatewiththesmallesttotalamountofnonproductivetimeisthenselectedforthesecondposition.Afterthebestfittingjobisaddedtothepartialsequence,thenewprofile(thedeparturetimesofthisjobfromallthemachines)iscomputedandtheprocedureisrepeated.FromtheremainingjobsintheMPSagaintheonethatfitsbestisselected.Thisprocesscontinuesuntilallthejobsarescheduled.ThePFheuristicfunctions,inasense,asadynamicdispatchingrule.ThePFheuristiccanbesummarizedasfollows.Algorithm6.2.2(ProfileFittingHeuristic).Step1.(InitialCondition)SelectthejobwiththelongesttotalprocessingtimeasthefirstjobintheMPS.Step2.(AnalysisofRemainingJobstobeScheduled)Foreachjobthathasnotyetbeenscheduleddothefollowing:Considerthatjobasthenextoneinthepartialsequenceandcomputethetotalnon-productivetimeonallmmachines(machineidletimeaswellasmachineblockingtime).Step3.(SelectionoftheNextJobinPartialSchedule)OfalljobsanalyzedinStep2,selectthejobwiththesmallesttotalnon-productivetimeasthenextoneinthepartialsequence.Step4.(StoppingCriterion)IfalljobsintheMPShavebeenscheduled,thenSTOP.Otherwise,gotoStep2.ObservethatinExample6.2.1,afterjob1,theProfileFittingheuristicwouldselectjob3togonext,asthiswouldcauseonlyoneunitofblockingtime(onmachine2)andnoidletimes.Ifjob2wereselectedtogoafterjob1,oneunitofidletimewouldbeincurredatmachine2andoneunitofblockingonmachine3,resultingintwounitsofnonproductivetime.Sointhisexampletheheuristicwouldyieldanoptimalsequence.ExperimentshaveshownthatthePFheuristicresultsingoodschedules.However,itcanberefinedtoperformevenbetter.Inthedescriptionabove,thegoodnessoffitofaparticularjobwasmeasuredbysummingallthenon-productivetimesonthemmachines.Eachmachinewasconsideredequallyimportant.Supposeonemachineisabottleneckwhichhasmoreprocess-ingtodothananyothermachine.Itisintuitivethatlosttimeonabottle-neckmachineisworsethanlosttimeonamachinethat,onaverage,doesnothavemuchprocessingtodo.Whenmeasuringthetotalamountoflost
6.2SequencingofUnpacedAssemblySystems123time,itmaybeappropriatetoweighteachinactivetimeperiodbyafac-torthatisproportionaltothedegreeofcongestionatthecorrespondingmachine.Thehigherthedegreeofcongestionatamachine,thelargertheweight.Onemeasureforthedegreeofcongestionofamachineiseasytocalculate;simplydeterminethetotalamountofprocessingtobedoneonalljobsinanMPSatthatmachine.Inthenumericalexamplepresentedabovethethirdandthefourthmachinesaremoreheavilyusedthanthefirstandsecondmachines(thesecondmachinewasnotusedatallandbasi-callyfunctionedasabuffer).Nonproductivetimeonthethirdandfourthmachinesisthereforelessdesirablethannonproductivetimeonthefirstandsecondmachines.Experimentshaveshownthatsuchaweightedver-sionofthePFheuristicperformssignificantlybetterthantheunweightedversion.Example6.2.3(ApplicationofWeightedProfileFittingHeuristic).ConsiderthreemachinesandanMPSoffourjobs.Therearenobuffersbe-tweenmachines.Theprocessingtimesofthefourjobsonthethreemachinesareasfollows.Jobs1234p1j2423p2j4402p3j2020Allthreemachinesareactualmachinesandnonearebuffers.Theworkloadsonthethreemachinesarenotentirelybalanced.Theworkloadonmachine1is11,onmachine2itis10andonmachine3itis4.So,timelostonmachine3islessdetrimentalthantimelostontheothertwomachines.ToapplyaweightedversionoftheProfileFittingheuristic,theweightgiventononproductivetimeonmachine3shouldbesmallerthantheweightsgiventononproductivetimesontheothertwomachines.Inthisexamplewesettheweightsformachines1and2equalto1andtheweightformachine3equalto0.Assumejob1isthefirstjobintheMPS.Theprofilecanbedeterminedeasily.Ifjob2isselectedtogosecond,thentherewillbenoidletimeorblockingonmachines1and2;howevermachine3willbeidlefor2timeunits.Ifjob3isselectedtogosecond,machines1and2arebothblockedfortwounitsoftime.Ifjob4isselectedtogosecond,machine1willbeblockedforoneunitoftime.Astheweightofmachine1is1andofmachine3is0,theweightedPFselectsjob2togosecond.ItcanbeverifiedthattheweightedPFresultsinthecycle1,2,4,3,withanMPScycletimeof12.IftheunweightedversionofthePFheuristicwereappliedandjob1againwasselectedtogofirst,thenjob4wouldbeselectedtogosecond.TheunweightedPFheuristicwouldresultinthesequence1,4,3,2withanMPScycletimeof14.
1246SchedulingofFlexibleAssemblySystems6.3SequencingofPacedAssemblySystemsInapacedassemblyline,aconveyorsystemmovesthejobs(e.g.,cars)fromoneworkstationtothenextataconstantspeed.Thejobsmaintainafixeddistancefromoneanother.Iftheassemblyataparticularstationisdonemanually,theworkerswalkalongsidethemovinglinewhileperformingtheiroperation;afteritscompletion,theywalkbacktowardsthatpointinthelinethatcorrespondstothebeginningoftheirstationorsection.Thetimeittakestowalkbackisoftennegligibleincomparisonwiththetimeittakestoperformtheoperation.Theoperationstobedoneatthevariousstationsare,ofcourse,differentandtheirprocessingtimesmaynotbeidentical.Thespacealongthelinereservedforaparticularoperationisinproportiontotheamountoftimeneededforthatoperation.Inthistypeofassemblylineastationisbasicallyequivalenttoadesignatedsectionoftheassemblyline.Atanypointintimetheremaybemorethanonejobatagivenstation;ifthatisthecase,thenseveraljobsmaybeprocessedatthatstationatthesametime.Inthisenvironmentbypassisnotallowed.Animportantcharacteristicofapacedassemblysystemisitsunitcycletime,whichisdefinedasthetimebetweentwosuccessivejobscomingofftheline.Theunitcycletimeisthereciprocaloftheproductionrate.Pacedassemblysystemsareverycommonintheautomotiveindustry.Assemblylinesintheautomotiveindustryhave,ofcourse,manyotherchar-acteristicsthatarenotincludedinthemodeldescribedabove.Oneoftheseis,forexample,thepointinthelinewheretheengineandthebodycometogether.Intheautomotiveindustrysomejobshaveduedates.Thesejobsaremadetoorderanditisimportanttoassembletheseinatimelyfashiontoprovidegoodcustomerservice.However,thesecommittedshippingdatesarenotasstringentasthoseinotherindustries(ajobmayhavetobeshippedinagivenweek,notnecessarilyonagivenday).Other(morestandard)jobsaremadeforinventoryandsenttodealers.Thetargetinventorylevelsofthedifferentmodelswiththevariousdifferentoptionpackagesaredeterminedbythemarketingdepartment.ThisimpliesthatproductionisamixtureofMake-To-OrderandMake-To-Stock.TheMake-To-Stockcomponentoftheproductionprovidessomeflexibilityfortheschedulingsystem,sincethetimingoftheproductionofthesejobsissomewhatflexible.Eachjobthatgoesthroughthelinehasanumberofattributesorcharac-teristicssuchascolor,options(e.g.,sunroof,powerwindows),anddestination.Fromaproductionpointofviewitisadvantageoustosequencejobsthathavecertainattributesincommononeafteranother.Forexample,itpaystohaveanumberofconsecutivejobswiththesamecolor,sincecleaningthepaintgunsinthepaintshopinvolvesachangeovercost.Also,ifanumberofjobshavetogoonthesametrailertothesamedestination,thenthesejobsshouldallappearwithinthesamesegmentofthesequence.Ifthefirstandthelastjobforoneparticulartraileraresequencedfarapart,thenthereisawaiting
6.3SequencingofPacedAssemblySystems125costinvolved.(Thedestinationsthatcauseproblemsareusuallythepointsoflowdemandwhereonlyasingletrailergoes.)Becauseofthecharacteris-ticsthatinvolvechangeovercosts,asequencingheuristichastogothroughagroupingphasewhereitattemptstokeepjobswithsimilarattributesto-gether.Groupingmayoccurwithregardtoalloperationsthathavesignificantsetuporchangeovercosts.Thisdoesnotincludeonlycoloranddestination,italsoincludesintricatemanualassembly,whereanimmediaterepetitionhasapositiveimpactonthequality.Thereisanotherclassofattributesthathaveadifferent(actually,anoppo-site)effectonthesequencing.Considertheinstallationofoptionalequipmentsuchasasunrooforthespecialpartsthatarerequiredinastationwagon.Agivenpercentageofthejobs,say10%,havetoundergosuchaspecialop-eration.Often,suchanoperationtakesmoretimethananoperationthatiscommontoalljobs;thesectionofthelineassignedtosuchanoperationmustthereforebelongerthanthesectionsofthelineassignedtootheroperations.Asthenumberofworkersassignedtosuchanoperation(e.g.,sunroofin-stallation)isproportionaltotheaverageworkload,thejobsthatneedthatoperationmustbespacedoutmoreorlessevenlyoverthesequence.Aheuris-tic,therefore,hastogothroughaspacingstepthatschedulessuchjobsatregularintervals.Example6.3.1(SunroofInstallation).Supposetheinstallationofasun-rooflastsfourtimeslongerthanatypicaloperationthathastobedoneonalljobs(e.g.,attachingthehood).Thesectionofthelinecorrespondingtothelongeroperationhastobeatleastfourtimeslongerthanthesectionofthelineassignedtotheshorteroperation.Ifthesectionofthelinecorrespondingtotheshorteroperationcontains,ontheaverage,asinglejob,thenthesec-tionofthelinecorrespondingtothelongeroperationcontains,ontheaverage,fourjobs.Ifonly10%ofthejobsneedtoundergothelongoperation,then,inaperfectlybalancedsequence,everytenthjobonthelinehastoundergothelongoperation.However,iftwoconsecutivejobsrequiretheinstallationofasunroof,thentheworkersmaynotbeabletocompletetheworkonthesecondjobintime.Thereasonisthefollowing:theycompletetheworkonthefirstjobwhenitisabouttoleavetheirsectionandthenturntowardsthenextjobandstartworkingonthatone.However,thissecondjobisalreadyrelativelyclosetotheendoftheirsectionanditmaynotbepossibletocom-pletetheoperationbeforeitleavestheirsection.Ifthesetwoparticularjobswerespacedmorethanfourpositionsapartinthesequence,thentherewouldnothavebeenanyproblem.Thistypeofoperationisusuallyreferredtoasacapacityconstrainedoperation.Foreachcapacityconstrainedoperationacriticalityindexcanbecomputed.Thiscriticalityindexistheminimumnumberofpositionsbetweentwojobsrequiringtheoperationdividedbytheaveragenumberofpositionsbetweentwosuchjobs.InExample6.3.1theindexis4/10.Thehighertheindex,themorecriticaltheoperation.Capacityconstrainedoperationsrequire
1266SchedulingofFlexibleAssemblySystemsacarefulbalancingofthesequence,sinceproperspacinghasasignificantimpactonquality.Thisspacingmustsatisfytheso-calledcapacityviolationconstraints;thesearebasicallyupperboundsonthefrequencieswithwhichjobsmayendupinpositionswherethecapacityconstrainedoperationscannotbecompletedintime.Basedontheconsiderationsdescribedabovetherearefourobjectivesthathavetobetakenintoaccountinthesequencingprocess.Oneimportantob-jectiveistheminimizationofthetotalsetupcost.Ifinthepaintshopajobwithcolorjisfollowedbyajobwithcolorkthereisasetupcostcjk.Iftwoconsecutivejobshavethesamecolor,thenthesetupcostiszero.Thefirstobjectiveistominimizethetotalsetupcostovertheentireproductionhorizon.AsecondobjectiveconcernstheduedatesoftheMake-To-Orderjobs.Ajob’sduedatecanbetranslatedintoacertainpositioninthesequence;ifthejobwouldappearafterits”duedateposition”itwouldbeconsideredlate.TheobjectivetominimizeissimilartothetotalweightedtardinessobjectivewjTj.Ifjobjistardy,itstardinessTjismeasuredbythenumberofposi-tionsinbetweenitsassignedpositionanditsduedateposition.Insomeplantsthisobjectiveisveryimportantwhileinotherplantsitisofnoimportance(e.g.,CadillacsaremoreoftenmadetoorderthanFordEscorts).Athirdobjectiveconcernsthespacingwithregardtothecapacitycon-strainedoperations.Letψi()denotethepenaltyincurredifworkstationihastodoworkontwojobsthatarepositionsapart.Thepenaltyψi()ismonotonicallydecreasinginandiszerowhenislargerthanagivenmini.Theobjectiveistominimizethesumofallψi().Thisthirdobjectivecouldbeconsideredaspartofamoregeneralobjectivethatisdescribednext.Afourthobjectiveconcernstherateofconsumptionofmaterialandpartsattheworkstations.Theobjectiveistokeeptheconsumptionratesofallthepartsatalltheworkstationsasregularaspossible.Therelativeweightsofthedifferentobjectivesdependonthespecificas-semblysystemandproductline.Insomeassemblylinesalljobsaremadetostockwhileinothersmostjobsaremadetoorder.Somelineshavenocapacityconstrainedoperations(suchassunroofinstallation)andthemaincostsaresetupcosts(e.g.,cleaningpaintguns).Oneprocedureforsequencingpacedassemblylinesistheso-calledGroup-ingandSpacing(GS)heuristic,whichhasbeenspecificallydesignedforthepacedassemblylinesthatarecommonintheautomotiveindustry.Inacaras-semblyfacilitythenumberofdifferentmodelsisusuallyfairlylargeandtherearemanydifferentoptionsoroptionpackagesthatacarcanhave.Beforetheheuristiccanbeappliedadetailedanalysishastobedoneofeachoperation.Operationswithsignificantsetupcostshavetoberankedindecreasingorderofsetupcostsandoperationswithcapacityconstraintshavetoberankedindecreasingorderoftheircriticalityindices.Theoutputofthisanalysisisaninputtotheheuristic.
6.3SequencingofPacedAssemblySystems127TheGSheuristicconsistsoffourphases.(i)Determiningthetotalnumberofjobstobescheduled.(ii)Groupingjobswithregardtooperationsthathavehighsetup-costs.(iii)Orderingofthesubgroupstakingshippingdatesintoaccount.(iv)Spacingjobswithinsubgroupstakingcapacityconstrainedoperationsintoaccount.Thefirstphasedeterminesthetotalnumberofjobstobescheduled.Thisnumberisusuallydecidedbytheschedulerbasedonhisknowledgeofthelikelihoodandthetimingofexogenousdisruptionsthatrequirerescheduling.Thisnumberisatrade-offbetweentwofactors.Ahighnumberallowstheschedulertofindasequencewithalowercost.However,theprobabilityofadisruptionishighandiftherearedisruptions,thentheproductionofsomejobsmaybepostponedunduly.Thenumbertobescheduledtypicallyrangesfromtheequivalentofonedayofproductiontooneweekofproduction.Thesecondphaseformssubgroupstakingoperationswithhighsetupcostsintoaccount.Therunlengthsofthedifferentcolorsaredetermined.Theserunlengthsmaybedifferentfordifferentcolors.Afrequentcolor,e.g.,grey,mayhavearunlengthofupto50,whereasaninfrequentcolor,e.g.,purple,mayhavearunlengthof1or2.Thegroupingwithregardtodestinationsisslightlydifferentsincetheconstraintsherearelessstrict.Jobswiththesamedestinationhavetobeclosetooneanotherinthesequence.However,jobsthathavetogotootherdestinationsmayalsobescatteredthroughoutthatsegmentofthesequence.Therunlengthsdeterminedinthisphasearetheresultsoftrade-offsbetweentheminimizationofsetup-costsandthemin-imizationoftheholdingcostsoffinishedjobs.TheyalsodependsomewhatonthecommittedshippingdatesoftheMake-To-Orderjobs.Assuch,deter-miningarunlengthisinasensesimilartocomputinganEconomicOrderQuantity(seeChapter7).Thethirdphaseordersthedifferentsubgroups.Theorderofthedifferentsubgroupsismainlydeterminedbytheurgencywithwhichthejobsinagrouphavetobeshipped.ThisurgencyisdeterminedbythecommittedshippingdatesoftheMake-To-OrderjobsaswellasthecurrentinventorylevelsoftheMake-To-Stockjobs.Thefourthphasedoestheinternalsequencingwithinthesubgroupscon-sideringthecapacityconstrainedoperations.Itfirstconsidersthemostcriticaloperationandspacesthejobsthathavetoundergothisoperationasuniformlyaspossible.Assumingthepositionsofthesejobsasfixed,itthenconsidersthesecondmostcapacityconstrainedoperationandspacesoutthesejobsasuniformlyaspossible.Beforepresentinganumericalexample,wedescribeasimplemathemat-icalmodelthathasmanyofthecharacteristicsofapacedassemblysystem.Considerasinglemachineandnjobs.Alltheprocessingtimesareequalto1(sincetheassemblylinemovesataconstantspeed).Jobjhasaduedatedjandaweightwj.Someduedatesmaybeinfinite.Jobjhaslparameters
1286SchedulingofFlexibleAssemblySystemsaj1,aj2,…,ajl.Thefirstparameterrepresentsthecolor,thesecondoneisequalto1whenthejobhasasunroofand0otherwise,thethirdonerepre-sentsthedestination,andsoon.Ifjobjisfollowedbyjobkandaj1=ak1(i.e.,thejobshavedifferentcolors),thenasetupcostcjkisincurred,whichisafunctionofaj1andak1.Ifbothjobsjandkhavesunroofs,i.e.,aj2=ak2=1,andtheyarespacedpositionsapart,apenaltyψ2()isincurred,whichisdecreasingin.Ifjobjiscompletedafteritsduedate,aweightedtardinesspenaltyisincurred.Theobjectiveistominimizethetotalcost,includingthesetupcosts,spacingcostsandtardinesscosts.Example6.3.2(ApplicationofGroupingandSpacingHeuristic).Considerasinglemachineand10jobs.Eachjobhasunitprocessingtime.Jobjhastwoparametersaj1andaj2.Iftwoconsecutivejobsjandkhaveaj1=ak1,thenasetupcostcjk=|aj1−ak1|isincurred.Iftwojobsjandkhaveaj2=ak2=1andtheyarespacedpositionsapart(i.e.,thereare−1jobsscheduledinbetween)apenaltycostψ2()=max(3−,0)isincurred.Somejobshaveduedates.Ifjobj,withduedatedj,iscompletedafteritsduedate,apenaltywjTjisincurred.Jobs12345678910aj11113335555aj20110111000dj∞2∞∞∞∞6∞∞∞wj0400004000Fromthedataitisclearthattherearethreegroupswithregardtotheoper-ationwithsetupcosts(e.g.,theremaybethreecolors:color1,color3,andcolor5).Theobjectiveistofindthesequencewiththeminimumtotalcost.ThefirstphaseoftheGSheuristicisnotapplicablehere.Therearethreegroupswithregardtoattribute1(color).GroupAconsistsofjobs1,2,and3,groupBconsistsofjobs4,5,and6,andgroupCconsistsofjobs7,8,9,and10.ThebestorderwithregardtosetupcostswouldbeA,B,C.However,groupCcontainsajobwithduedate6thatwouldnotbecompletedintimeifthegroupsareorderedthatway.Sincethetardinesspenaltyissomewhathigh,itmaybebettertoorderthegroupsA,C,B,becauseinthiswayjob7canbecompletedintime.SotheresultofthethirdphaseoftheGSheuristicisthatthegroupsareorderedinthesequenceA,C,B.ThelastphaseoftheGSheuristicconsidersthecapacityconstrainedoper-ationswhichareembodiedinattribute2ofthismodel.Thejobswithattribute2equalto1havetobespacedoutasmuchaspossible.Itcanbeverifiedthat
6.4SchedulingofFlexibleFlowSystemswithBypass129thefollowingsequenceminimizesthepenaltieswithregardtothecapacityconstrainedoperation:GroupAintheorder2,1,3,followedbyGroupCintheorder8,7,9,10,andGroupBintheorder5,4,6.Thesequenceofat-tribute2valuesisthen1,0,1,0,1,0,0,1,0,1.Thetotalcostwithregardstothecapacityconstrainedoperationis3.Thus,thetotalcostassociatedwiththissequenceis6+3=9(6becauseofsetupcostand3becauseofthecapacityconstrainedoperation).Theexamplepresentedaboveisclearlyverysimple.Itwasnotnecessarytodeterminethesizesofeachgroup,sinceitappearedthattherewouldbejustasinglegroupforeachvalueofattribute1.Inlargerinstanceswithmorejobs,thequestionofgroupsizesbecomes,ofcourse,amoreimportantissue.Itisnoteasytoformulateaprecisemodelandalgorithmforthepacedassemblylinesequencingproblemingeneral.Thedifficultyliesinthevariousobjectivefunctions.Thestructureofanalgorithmdependsontherelativeimportanceofeachoneoftheobjectives.Mostpacedassemblysystemsintherealworldaresignificantlymorecom-plicatedthanthosedescribedinthissection.Forexample,pacedassemblylinesintheautomotiveindustrymayhavearesequencingbankimmediatelyafterthepaintshopthatbasicallypartitionstheoverallproblemintotwomoreorlessindependentsubproblems.Anotherfactoristhatsomecarsmayneedtwopassesthroughthepaintshop.Thegroupingandspacingheuris-ticsthathavebeenimplementedinindustryare,therefore,significantlymorecomplicatedthanthesimpleproceduredescribedabove.Pacedassemblysequencingproblemshavealsobeentackledwithcon-straintprogrammingtechniques.AppendixDdescribesaprogramforapacedassemblysystemthatisencodedintheconstraintprogramminglanguageOPL.6.4SchedulingofFlexibleFlowSystemswithBypassConsideranassemblysystemwithanumberofstagesinseriesand,ateachstage,anumberofmachinesinparallel.Ajob,whichinthiscaseoftenisequiv-alenttoabatchofidenticalitemssuchasPrintedCircuitBoards(PCB’s),needsprocessingateverystage,butonlyononemachine.Usually,anyofthemachineswilldo,butitmaybethecasethatatagivenstagenotallmachinesareidenticalandthatagivenjobhastobeprocessedonacertainmachine.Ifajobdoesnotneedprocessingatastage,thematerialhandlingsystemwillallowthatjobtobypassthatstageandallthejobsresidingthere,seeFigure6.2.Abufferatastagemayhavealimitedcapacityandwhenabufferisfull,theneitherthematerialhandlingsystemmustcometoastandstill,or,ifthereisanoptiontorecirculate,thejobsmustbypassthatstageandrecirculate.Themanufacturingprocessisrepetitiveandthereforeitisofinteresttofindagoodcyclicschedule(similartotheoneinSection6.2).
1306SchedulingofFlexibleAssemblySystemsStage 1Material-handlingsystemStage 2Fig.6.2.FlexibleflowlinewithbypassTheFlexibleFlowLineLoading(FFLL)algorithmwasdesignedatIBMforthemachineenvironmentdescribedabove.ItwasoriginallyconceivedforanassemblysystemusedfortheinsertionofcomponentsinPCB’s.ThetwomainobjectivesofthealgorithmarethemaximizationofthroughputandtheminimizationofWIP.Withthegoalofmaximizingthethroughputanattemptismadetominimizethemakespanofawholeday’smix.TheFFLLalgorithmactuallyattemptstominimizethecycletimeofaMinimumPartSet(MPS).(ForthedefinitionoftheMPS,seeSection6.2.)Asbufferspacesarelimited,ittriestominimizetheWIPtoreduceblockingprobabilities.TheFFLLalgorithmconsistsofthreephases:(i)Themachineallocationphase.(ii)Thesequencingphase.(iii)Thereleasetimingphase.Themachineallocationphaseassignseachjobtoaparticularmachineineachbankofmachines.Machineallocationisdonebeforesequencingandtiming,because,inordertoperformthelasttwophases,theworkloadassignedtoeachmachinemustbeknown.Thelowestconceivablemaximumworkloadforabankwouldbeobtainedifallthemachinesinabankweregivenanequalworkload.Inordertoobtainnearlybalancedworkloadsforthemachinesatabank,theLongestProcessingTimefirst(LPT)heuristicisused(seeSection5.2).Inthisheuristicallthejobsare,forthetimebeing,assumedtobeavailableatthesametimeandareallocatedoneatatimetothenextavailablemachineindecreasingorderoftheirprocessingtimes.Aftertheallocationisdeterminedinthisway,theitemsassignedtoamachinemayberesequenced;thisdoesnotaltertheworkloadbalanceoverthemachinesatagivenbank.Theoutputofthisphaseismerelytheallocationofjobstomachinesandnotthesequencingofthejobsorthetimingoftheirprocessing.ThesequencingphasedeterminestheorderinwhichthejobsoftheMPSarereleasedintothesystem.ThishasasignificantimpactontheMPScycle
6.4SchedulingofFlexibleFlowSystemswithBypass131time.TheFFLLalgorithmusesaso-calledDynamicBalancingheuristicforsequencinganMPS.Thisheuristicisbasedontheintuitionthatjobstendtoqueueupinthebufferofamachinewhenalargeworkloadissenttothatmachineinashortperiodoftime.Thisoccurswhenthereisanintervalintheloadingsequencethatcontainsmanyjobswithlongprocessingtimesallocatedtoonemachine.LetnbethenumberofjobsinanMPSandmthenumberofmachinesintheentiresystem.Letpijdenotetheprocessingtimeofjobjonmachinei.Notethatpij=0forallbutonemachineinabank.LetWi=nj=1pijdenotetheworkloadinanMPSthatisassignedtomachinei,andletW=mi=1WidenotethetotalworkloadofanMPS.Assumingafixedsequence,letJkdenotethesetofjobsloadedintothesystemuptoandincludingjobk,andletαik=j∈JkpijWi.Theαikrepresentthefractionofthetotalworkloadofmachineithathasenteredthesystembythetimejobkisloaded.Clearly,0≤αik≤1.TheDynamicBalancingprocedureattemptstokeeptheα1k,α2k,…,αmkasclosetooneanotheraspossible,i.e.,asclosetoanidealtargetα∗k,thatisdefinedasα∗k=j∈Jkmi=1pijnj=1mi=1pij=j∈Jkpj/W,wherepj=mi=1pijistheworkloadontheentiresystemduetojobj.Henceα∗kisthefractionofthetotalsystemworkloadthatisreleasedintothesystembythetimejobkisloaded.Thecumulativeworkloadonmachinei,i.e.,j∈Jkpij,shouldbeclosetothetargetα∗kWi.Now,letoikdenoteameasureofoverloadatmachineiduetojobkenteringthesystem,definedasoik=pik−pkWi/W.
1326SchedulingofFlexibleAssemblySystemsClearly,oikmaybenegative,whichimpliesanunderload.LetOik=j∈Jkoij=j∈Jkpij−α∗kWidenotethecumulativeoverload(orunderload)onmachineiduetothejobsinthesequenceuptoandincludingjobk.Tobeexactlyontargetmeansthatmachineiisneitheroverloadednorunderloadedwhenjobkentersthesystem,i.e.,Oik=0.TheDynamicBalancingheuristicattemptstominimizenk=1mi=1max(Oik,0).Theprocedureisbasicallyagreedyheuristic,whichselectsfromamongtheremainingitemsintheMPStheonethatminimizestheobjectiveatthatpointinthesequence.Thereleasetimingphaseworksasfollows.FromtheallocationphasetheMPSworkloadateachmachineisknown.ThemachinewiththegreatestMPSworkloadisthebottlenecksincetheMPScycletimecannotbesmallerthantheMPSworkloadatthebottleneckmachine.WewishtodetermineatimingmechanismthatyieldsaschedulewithaminimumMPScycletime.First,letalljobsintheMPSenterthesystemasrapidlyaspossible.Considerthemachinesoneatatime.Ateachmachinethejobsareprocessedintheorderinwhichtheyarriveandprocessingstartsassoonasthejobisavailable.Thereleasetimesarenowmodifiedasfollows.Assumethatthestartingandcom-pletiontimesatthebottleneckmachinearefixed.First,considerthemachinesthatareupstreamofthebottleneckmachineanddelaytheprocessingofalljobsoneachofthesemachines,asmuchaspossible,withoutalteringthejobsequences.Thedelaysarethusdeterminedbythestartingtimesatthebottle-neckmachine.Second,considerthemachinesthatarepositioneddownstreamfromthebottleneckmachine.Processalljobsonthesemachinesasearlyaspossible,againwithoutalteringjobsequences.Thesemodificationsinreleasetimestendtoreducethenumberofjobswaitingforprocessing,thusreducingrequiredbufferspace.Thisthreephaseprocedureattemptstofindthecyclicschedulewithmin-imumMPScycletimeinasteadystate.Ifthesystemstartsoutemptyatsomepointintime,itmaytakeafewMPS’storeachasteadystate.Usually,thistransientperiodisveryshort.Thealgorithmtendstoachieveshortcycletimesduringthetransientperiodaswell.ExtensiveexperimentswiththeFFLLalgorithmindicatesthatthemethodisavaluabletoolfortheschedulingofflexibleflowlines.Example6.4.1(ApplicationofFFLLAlgorithm).Consideraflexibleflowshopwiththreestages.Atstages1and3therearetwomachinesinparallel.Atstage2,thereisasinglemachine.TherearefivejobsinanMPS.Letphjdenotetheprocessingtimeofjobjatstageh,h=1,2,3.
6.4SchedulingofFlexibleFlowSystemswithBypass133Jobs12345p1j63135p2j32132p3j45634ThefirstphaseoftheFFLLalgorithmperformsanallocationprocedureforstages1and3.ApplyingtheLPTheuristictothefivejobsonthetwomachinesinstage1resultsinanallocationofjobs1and4toonemachineandjobs5,2and3totheothermachine.Bothmachineshavetoperformatotalof9timeunitsofprocessing.ApplyingLPTtothefivejobsonthetwomachinesinstage3resultsinanallocationofjobs3and5toonemachineandjobs2,1and4totheothermachine(inthiscaseLPTdoesnotyieldanoptimalbalance).Notethatmachines1and2areatstage1,machine3isatstage2andmachines4and5areatstage3.Ifpijdenotestheprocessingtimeofjobjonmachinei,wehaveJobs12345p1j60030p2j03105——-p3j32132——-p4j45030p5j00604TheworkloadWiofmachineiduetoasingleMPScannowbecomputed.TheworkloadvectorWiis(9,9,11,12,10)andtheentireworkloadWis51.Theworkloadimposedontheentiresystemduetojobj,pj,canalsobecomputed.Thepjvectoris(13,10,8,9,11).Basedonthesenumbers,allvaluesofoikcanbedetermined,e.g.,o11=6−13×9/51=+3.71o21=0−13×9/51=−2.29o31=3−13×11/51=+0.20o41=4−13×12/51=+0.94o51=0−13×10/51=−2.55Computingtheentireoikmatrixyields:+3.71−1.76−1.41+1.41−1.94−2.29+1.23−0.41−1.59+3.06+0.20−0.16−0.73+1.06−0.37+0.94+2.64−1.88+0.88−2.59−2.55−1.96+4.43−1.76+1.84
1346SchedulingofFlexibleAssemblySystemsOfcourse,thesolutionalsodependsontheinitialjobintheMPS.IftheinitialjobischosenaccordingtotheDynamicBalancingheuristic,thenjob4goesfirstandOi4=(+1.41,−1.59,+1.06,+0.88,−1.76).Therearefourjobsthatqualifytogosecond,namelyjobs1,2,3and5.Ifjobkgoessecond,therespectiveOikvectorsare:Oi1=(+5.11,−3.88,+1.26,+1.82,−4.31)Oi2=(−0.35,−0.36,+0.90,+3.52,−3.72)Oi3=(+0.00,−2.00,+0.33,−1.00,+2.67)Oi5=(−0.53,+1.47,+0.69,−1.71,+0.08)Thedynamicbalancingheuristicthenselectsjob5togosecond.Proceedinginthesamemannerthedynamicbalancingheuristicselectsjob1togothirdandOi1=(+3.18,−0.82,+0.89,−0.77,−2.47).Proceeding,weseethatjob3goesfourth.ThenOi3=(+1.76,−1.23,+0.16,−2.64,+1.96).Thefinalcycleisthus4,5,1,3,2.Applyingthereleasetimingphasetothiscycleresultsinitiallyinthesched-ulepresentedinFigure6.3.TheMPScycletimeof12isactuallydeterminedbymachine4(thebottleneckmachine)andthereisthereforenoidletimeallowedbetweentheprocessingofjobsonthismachine.Itisclearthattheprocessingofjobs3and2onmachine5canbepostponedbythreetimeunits.Themodeldiscussedinthissectioncanbecomparedtoaflexibleflowshop(whichisaspecialcaseofaflexiblejobshop).TherearesimilaritiesaswellasdifferencesbetweenthemodelconsideredinthissectionandaflexibleflowshopthatfitswithintheframeworkofChapter5.Themachineenvironmentsinthetwomodelsaresimilar,sincebothsettingsareflexibleflowshopsthatallowbypass.Inthemodeldescribedinthissectionthereisthematerialhandlingsystemthatimposesadditionalconstraintsthathaveanimpactontheobjectives.ThelimitedbuffersintheflexibleassemblysystemrequireaminimizationoftheWork-In-Process.Theobjectivesinthetwomodelsarealsodifferentfromanotherpointofview.ThemodelsdescribedinChapter5tendtobemoreMake-To-Order;animportantobjectiveinsuchmodelsmaybetheminimizationofthetotalweightedtardiness.ThemodelconsideredinthissectionisbasicallyaMake-To-Stockmodel.Themainobjectiveisthemaximizationofthroughput.6.5MixedModelAssemblySequencingatToyotaAmongthemajorcarmanufacturersToyotahasbeenalwaysoneofthemoreinnovativeonesasfarasmanufacturingandassemblyisconcerned.Toyota
6.5MixedModelAssemblySequencingatToyota135Machine 1Machine 2Machine 3Machine 4Machine 50102030t0102030tMachine 1Machine 2Machine 3Machine 4Machine 5111114444455555333355333222211445533222114422235141414532235Fig.6.3.GanttchartsfortheFFLLalgorithmoperatesaccordingtotheJust-In-Time(JIT)principle.TomaketheJIToperationrunsmoothly,itisimportantthattheconsumptionratesofallthepartsateachstationarekeptasregularaspossible.Toyota’smostimportantgoalintheoperationofitsmixedmodelassemblylinesistokeeptheratesofconsumptionofallpartsmoreorlessconstant.Inotherwords,thequantityofeachpartusedperhourmustbemaintainedasregularaspossible.Asdescribedinthesectiononpacedassemblysystems,balancingtheworkloadateachoneoftheworkstationsovertimeisoftenanimportantobjectiveincarassembly.Somecarsmayneedmorethantheaverageamountofprocessingataparticularworkstation,whileothersmayneedless.However,ifToyotamanagestokeeptheconsumptionofallthepartsateachoneoftheworkstationsmoreorlessconstant,thentheworkloadsarebalancedaswell.InordertoformalizetheToyotaobjective,letNdenotethetotalnumberofcarstobesequenced.Thisnumberisameasureoftheplanninghorizon.LetdenotethenumberofdifferentmodelsandNj,j=1,…,,thenumberof
1366SchedulingofFlexibleAssemblySystemsunitsofmodeljthathavetobeproducedovertheperiodunderconsideration.ThusN=j=1Nj.Inpracticetheplannedproductionquantitymaybearound500andthenum-berofdifferentmodelsaround180.LetνdenotethenumberofdifferenttypesofpartsneededbyallworkstationsfortheassemblyoftheNcarsandletbij,i=1,…,ν,j=1,…,,denotethenumberofpartsoftypeineededfortheassemblyofoneunitofmodelj.LetRidenotethetotalnumberofpartsoftypeirequiredfortheassemblyofallNcarsandxikthetotalnumberofpartsoftypeineededintheassemblyofthefirstkunitsinthesequence.SoRi/NistheaveragenumberofpartirequiredfortheassemblyofacarandkRi/Ntheaveragenumberofpartirequiredfortheassemblyofkcars.Tokeeptherateofconsumptionofpartiasregularaspossible,thevariablexikshouldbeascloseaspossibletothevaluekRi/N.ConsidernowallνpartsandthetwovectorskR1N,…,kRνNand(x1k,…,xνk).Thefirstvectorplaystheroleofthetargetorgoal,whilethesecondvectorrepresentstheactualnumbersofpartsneededfortheassemblyofthefirstkunitsinthegivensequence.Let∆k=νi=1kRiN−xik2denoteameasureofthedifferencebetweenthetwovectors.Let∆k(j)denotethevalueofthisdifferenceifmodeljhasbeenputinthek-thposition,i.e.,∆k(j)=νi=1kRiN−xi,k−1−bij2.Inordertominimizethisobjective,ToyotadevelopedtheGoalChasingmethodwhichcanbedescribedasfollows.Algorithm6.5.1(GoalChasingMethod).Step1.Setxi0=0,S0={1,2,…,},andk=1.Step2.Selectforthek-thepositioninthesequencemodelj∗thatminimizesthemeasure∆k(j),i.e.,
6.6Discussion137∆k(j∗)=minj∈Sk−1νi=1kRiN−xi,k−1−bij2,Step3.Ifmoreunitsofmodelj∗remaintobesequenced,setSk=Sk−1Ifallunitsofmodelj∗havenowbeensequenced,setSk=Sk−1−{j∗}.Step4.IfSk=∅,thenSTOP.IfSk=∅,setxik=xi,k−1+bij∗,i=1,…,ν.Setk=k+1andgotoStep2.Sincethenumberofpartsinacarisaround20,000,itisinpracticediffi-culttoapplytheGoalChasingmethodtoallparts.Therefore,thepartsarerepresentedonlybytheirrespectivesubassembly.Thenumberofsubassem-bliesisaround20andToyotagivestheimportantsubassembliesadditionalweight.Thesubassembliesinclude:(i)engines,(vi)bumpers,(ii)transmissions,(vii)steeringassemblies,(iii)frames,(viii)wheels,(iv)frontaxles,(ix)doors,and(v)rearaxles,(x)airconditioners.ThefactthattheconsumptionratesofthepartsarethemainobjectivegivesanindicationofhowimportantJITisforToyota.NotethesimilaritybetweentheGoalChasingmethodandtheDynamicBalancingroutinedescribedinSection6.4.ThephysicalenvironmentatToy-otais,ofcourse,moresimilartotheenvironmentofthepacedassemblysys-temsdescribedinSection6.3.6.6DiscussionSchedulingproblemsinflexibleassemblysystemsusuallyarenotaseasytoformulateasthemoreconventionaljobshopschedulingproblems.Thema-terialhandlingsystemsorconveyorsystemsimposeconstraintsthattendtobeapplication-specificanddifficulttoformulatemathematically.Sincetheseproblemsarehardtoformulateasmathematicalprogramsthesolutionpro-ceduresareusuallynotbasedonbranch-and-bound,butratherbasedonheuristics.Theframeworkofaheuristicistypicallymodularandapplication-specific.Otherapproachesincludeconstraintprogrammingtechniques.Ap-pendixDpresentsaconstraintprogrammingapplicationofamixedmodelassemblyline.
1386SchedulingofFlexibleAssemblySystemsAnotherclassofsystemsthatsharemanyoftheschedulingcomplexitiesofflexibleassemblysystemsaretheflexiblemanufacturingsystems.Ingeneral,aflexiblemanufacturingsystemmaybedefinedasaproductionsystemthatiscapableofproducingavarietyofparttypes;itconsistsof(numericallycon-trolled)machinesthatareconnectedtooneanotherbyanautomatedmaterialhandlingsystem.Theentiresystemtypicallyisundercomputercontrol.Withtheappropriatetoolsetups,eachmachineinthesystemcanperformdifferentoperations.Anoperationmaythereforebeperformedatanyoneofanumberofmachines.Theroutingofajobthroughthesystemisthereforeflexibleandisatypeofdecisionthathastobemadeintheschedulingprocess.Twoothertypesofdecisionsarethesequencingoftheoperationsonthemachinesandthesetupsofthetoolsonthemachines.Theseschedulingproblemsareharderthantheclassicaljobshopschedulingproblemsandtheschedulingproblemsthatoccurinflexibleassemblysystems.Asinflexibleassemblysystems,ma-terialhandlingsystemsandlimitedbuffersincreasethecomplexity.Justasaflexibleassemblysystemoftencanbeviewedasa(flexible)flowshopwithanumberofadditionalconstraints,aflexiblemanufacturingsystemcanbeviewedasa(flexible)jobshopwithanumberofadditionalconstraints.Whilealimitedbufferinaflexibleassemblysystemmaycauseblocking,alimitedbufferinaflexiblemanufacturingsystemmaynotonlycauseblockingbutalsodeadlock.Theschedulingproblemsinflexiblemanufacturingsystemsarenoteasytoformulateasmathematicalprogramswhenalldecisionsandconstraintshavetobeincludedintheformulation.Evenwhentheycanbeformulatedasmathematicalprograms,theyarestillveryhardtosolve.Manyformulationsthereforeincorporateonlyoneortwoofthethreetypesofdecisionslistedaboveandignoresomeoftheresourceconstraints.Mostoftheresearchinflexibleassemblysystemsandflexiblemanufac-turingsystemshavefocusedonspecificproblemsarisinginindustry.Whileaclassificationschemehasbeenavailableforjobshopschedulingproblemsforquitesometime,itisonlyrecentlythatsuchaschemehasbeenemergingforschedulingproblemsinflexiblemanufacturingsystems.Exercises6.1.ConsiderinthemodelofSection6.2anMPSof5jobs.(a)Showthatwhenalljobsaredifferentthenumberofdifferentcyclicschedulesis4!.(b)Computethenumberofdifferentcyclicscheduleswhentwojobsarethesame(i.e.,therearefourdifferentjobtypesamongthe5jobs).6.2.ConsiderthemodeldiscussedinSection6.2with4machinesandanMPSof4jobs.
Exercises139Jobs1234p1j6468p2j21046p3j4802p4j8266(a)ApplytheunweightedPFheuristictofindacyclicschedule.Choosejob1astheinitialjobandcomputetheMPScycletime.(b)ApplyagaintheunweightedPFheuristic.Choosejob2astheinitialjobandcomputetheMPScycletime.(c)Findtheoptimalschedule.6.3.Considerthesameproblemasinthepreviousexercise.(a)ApplyaweightedPFheuristictofindacyclicschedule.Choosetheweightsassociatedwithmachines1,2,3,4as2,2,1,2,respectively.Selectjob1astheinitialjob.(b)ApplyagainaweightedPFheuristicbutnowwithweights3,3,1,3.Selectagainjob1astheinitialjob.(c)Repeatagain(a)and(b)butselectjob2astheinitialjob.(d)Comparetheimpactoftheweightsontheheuristic’sperformancewiththeimpactoftheselectionofthefirstjobontheperformance.6.4.ConsiderthemodeldiscussedinSection6.2.Assumethatasystemisinasteadystateifeachmachineisinasteadystate.Thatis,ateachmachinethedepartureofjobjinoneMPSoccursexactlythecycletimebeforethedepartureofjobjinthenextMPS.Constructanexamplewith3machinesandanMPSofasinglejobthattakesmorethan100MPS’storeachasteadystate,assumingthesystemstartsoutempty.6.5.Inordertomodelapacedassemblylineconsiderasinglemachineandnjobs.Theprocessingtimesofeachoneofthejobsis1.Jobjhasaduedatedjandaweightwj.Ifjobjiscompletedafteritsduedate,thenapenaltywjTjisincurred.Therearesequencedependentsetupcostscjkbutnosetuptimes.Thesequencedependentsetupcostsaredeterminedasfollows:jobjhasanattributeajthatcanbeeither0or1.Mostjobshaveanattributevalueajequalto0.Iftherearetwojobswithajequalto1sequencedinsuchawaythattherearelessthan3jobswithajequalto0inbetween,thenapenaltycostc1=3isincurred.(Thisimpliesthatthisattributeajcorrespondstoacapacityconstrainedoperation).(a)Designanalgorithmforfindingasequencewithminimumcost.(Hint:Consider,forexample,acompositedispatchingrulefollowedbyalocalsearch;seeAppendixC.)(b)Applythealgorithmdevelopedtothefollowinginstance.
1406SchedulingofFlexibleAssemblySystemsJobs12345678910aj0110001000dj∞2∞1∞56∞2∞wj0401034030Computethetotalcostofthesequence.6.6.ApplytheGSheuristictotheinstanceinExercise6.5.Comparetheresultwiththatobtainedinthepreviousexercise.6.7.ConsiderthemodelinExercise6.5.Nowjobjhastwoattributesajandbj.TheajattributeisthesameasinExercise6.5.Thebjvaluesalsocanbeeither0or1.Iftwojobswithbjvalueequalto1arepositionedinsuchawaythatthereare4orlessjobswithbjvalueequalto0inbetween,thenapenaltycostc2=3isincurred.(Thissituationcorrespondstoalinewithtwocapacityconstrainedoperations).(a)DescribehowthealgorithmofExercise6.5hastobemodifiedtotakethisgeneralizationintoaccount.(b)Applyyouralgorithmtotheinstancebelow.Jobs12345678910aj0110001000bj0101101000dj∞2∞1∞56∞2∞wj0401034030Computethetotalcostofthesequence.6.8.ConsidertheapplicationoftheFFLLalgorithminExample6.4.1.Insteadoflettingthedynamicbalancingheuristicminimizemi=1nk=1max(Oik,0),letitminimizemi=1nk=1|Oik|.RedoExample6.4.1andcomparetheperformancesofthetwodynamicbal-ancingheuristics.6.9.ConsidertheapplicationoftheFFLLalgorithmtotheinstanceinEx-ample6.4.1.InsteadofapplyingLPTinthefirstphaseofthealgorithm,findtheoptimalallocationofjobstomachines(whichleadstoaperfectbalanceofmachines4and5).Proceedwiththesequencingphaseandreleasetimingphasebasedonthisnewallocation.
CommentsandReferences1416.10.ConsidertheinstanceinExample6.4.1again.(a)ComputeinExample6.4.1thenumberofjobswaitingforprocessingateachstageasafunctionoftimeanddeterminetherequiredbuffersizeateachstage.(b)ConsidertheapplicationoftheFFLLalgorithmtotheinstanceinEx-ample6.4.1withthemachineallocationasprescribedinExercise6.9.Com-putethenumberofjobswaitingforprocessingateachstageasafunctionoftimeanddeterminetherequiredbuffersize.(Notethatwithregardtothemachinesbeforethebottleneck,thereleasetimingphaseinasensepostponesthereleaseofeachjobasmuchaspossibleandtendstoreducethenumberofjobswaitingforprocessingateachstage.)CommentsandReferencesFlexibleassemblysystemsconstituteasubcategoryofflexiblemanufacturingsys-tems(FMS).(AnothersubcategoryofFMSarethegeneralflexiblemachiningsys-tems(GFMS).)AsignificantamountofresearchhasbeendoneonschedulinginFMSingeneral.ForanoverviewofschedulingresearchinFMS,seeMacCarthyandLiu(1993),RachamaduguandStecke(1994),andBasnetandMize(1994).LiuandMacCarthy(1996)clarifythebasicconceptsofschedulinginFMSandclassifytheseschedulingproblemsaccordingtoaclassificationschemethatisbasedontheconfigurationsofthesesystems.Foracomprehensivemixedintegerlinearprogram-mingmodelforthebasicschedulingproblemthatoccursinanFMS,seeLiuandMacCarthy(1997).TheschedulingofasingleflexiblemachineisstudiedbyTangandDenardo(1988).Acertainamountofresearchhasbeendoneontheschedulingofflexibleas-semblysystems.TheunpacedassemblysystemwithlimitedbuffersisanalyzedbyPinedo,WolfandMcCormick(1986),McCormick,Pinedo,ShenkerandWolf(1989),andAbadie,HallandSriskandarajah(2000).ItstransientanalysisisdiscussedinMcCormick,Pinedo,ShenkerandWolf(1990).Formoreresultsoncyclicscheduling,seeMatsuo(1990)andRoundy(1992),andformoreresultsonflowshopswithlim-itedbuffers,seeWismer(1972)andPinedo(1982).HallandSriskandarajah(1996)presentasurveythatincludesmanyofthesecyclicschedulingmodelswithlimitedbuffers.Fortheschedulingofaroboticassemblysystem,seeHall,KamounandSriskandarajah(1997).ThebookbyScholl(1998)focusesonthebalancingandsequencingofassem-blylines(pacedassemblysystems).Forschedulingproblemsandsolutionsintheautomotiveindustry,see,forexample,BurnsandDaganzo(1987),YanoandBolat(1989),Bean,Birge,MittenthalandNoon(1991),Garcia-Sabater(2001)andDrexlandKimms(2001).Importantaspectsofschedulingproblemsintheautomotiveindustryarethesequencedependentsetupcostsandsetuptimes.TheCD-ROMthatisattachedtothisbookcontainsseveralmini-casesthatfocusonassemblylinesequencingintheautomotiveindustry(projectsatNissanandPeugeot).Aconsiderableamountofresearchhasfocusedspecificallyonsequencedepen-dentsetups;see,forexample,GilmoreandGomory(1964),andBianco,Ricciardelli,RinaldiandSassano(1988).Crama,KolenandOerlemans(1990)developacompre-
1426SchedulingofFlexibleAssemblySystemshensivehierarchicaldecisionmodelforplanningamultiplemachineflowlinewithsequencedependentsetups.TheschedulingproblemconcerningtheflexibleflowlinewithlimitedbuffersandbypassisbasedonasettingfoundatIBMandanalyzedbyWittrock(1985,1988,1990).ThewayToyotaschedulesitsassemblylinesisdiscussedinMonden(1983);Section6.5isbasedonMonden’sAppendix2.
Chapter7EconomicLotScheduling7.1Introduction……………………………1437.2OneTypeofItemandtheEconomicLotSize…1447.3DifferentTypesofItems-RotationSchedules..1487.4DifferentTypesofItems-ArbitrarySchedules.1527.5MoreGeneralELSPModels……………….1617.6MultiproductPlanningandSchedulingatOwens-CorningFiberglas………………….1647.7Discussion……………………………..1667.1IntroductionInajobshop,eachjobhasitsownidentityanditsownsetofprocessingrequirements.Inaflexibleassemblysystem,thereareanumberofdifferenttypesofjobsandjobsofthesametypehaveidenticalprocessingrequirements;insuchasystem,setuptimesandsetupcostsareoftennotimportantandaschedulemayalternatemanytimesbetweenjobsofdifferenttypes.Inaflexibleassemblysystemanalternatingscheduleisoftenmoreefficientthanaschedulewithlongrunsofidenticaljobs.Inthemodelsconsideredinthischapter,asetofidenticaljobsmaybelargeandsetuptimesandsetupcostsbetweenjobsoftwodifferenttypesmaybesignificant.Asetuptypicallydependsonthecharacteristicsofthejobabouttobestartedandtheonejustcompleted.Ifajob’sprocessingonamachinerequiresamajorsetupthenitmaybeadvantageoustoletthisjobbefollowedbyanumberofjobsofthesametype.Inthischapterwerefertojobsasitemsandwecalltheuninterruptedprocessingofaseriesofidenticalitemsarun.Ifafacilityormachineisgearedtoproduceidenticalitemsinlongruns,thentheproductiontendstobeMake-To-Stock,whichinevitablyinvolvesinventoryholdingcosts.This© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_7,143
1447EconomicLotSchedulingformofproductionis,attimes,alsoreferredtoascontinuousmanufacturing(incontrasttotheformsofdiscretemanufacturingconsideredinthepreviouschapters).Thetimehorizonincontinuousmanufacturingisoftenintheorderofmonthsorevenyears.Theobjectiveistominimizethetotalcost,whichincludesinventoryholdingcostaswellassetupcost.Theoptimalscheduleistypicallyatrade-offbetweeninventoryholdingcostsandsetupcostsandisoftenrepetitiveorcyclic.Theassociatedschedulingproblemhasseveralaspects.First,thelengthsoftherunshavetobedeterminedand,second,theorderofthedifferentrunshastobeestablished.Therunlengthsaretypicallyreferredtoasthelotsizesandtheyaretheresultoftrade-offsbetweensetupcostsandinventoryholdingcosts.Thelotshavetobesequencedinsuchawaythatthesetuptimesandsetupcostsareminimized.ThisschedulingproblemisreferredtoastheEconomicLotSchedulingProblem(ELSP).InthestandardELSPasinglefacilityormachinehastoproducendifferentitems.ThemachinecanproduceitemsoftypejatarateofQjperunittime.Ifanitemoftypejisregardedasajobwithprocessingtimepj,thenQj=1/pj.WeassumethatthedemandratefortypejisconstantatDjitemsperunittime.Theinventoryholdingcostforoneitemoftypejishjdollarsperunittime.Ifanitemoftypejisfollowedbyanitemoftypekasetupcostcjkisincurred;moreover,asetuptimesjkmayberequired.Insomemodelsweassumethatasetupinvolvesacostbutnomachinetimeandinother,moregeneral,modelsweassumethatasetupinvolvesacostaswellasmachinetime.Thesetupcostandtimemaybeeithersequencedependentorindependent.Ifthesetupcost(time)issequenceindependent,thencjk=ck(sjk=sk).Theproblemcanbeviewedasoneofdecidingacyclelengthxandasequenceofrunsorcyclej1,j2,…,jν.Thissequencemaycontainrepetitions,soν≥n.Theassociatedruntimesareτj1,τj2,…,τjνandtheremaybeidletimebetweentwoconsecutiveruns.Inpractice,therearemanyapplicationsofeconomiclotscheduling.Intheprocessindustries(e.g.,thechemical,paper,pharmaceutical,aluminumandsteelindustries)setupcostsandinventoryholdingcostsaresignificant.Whenminimizingthetotalcosts,aschedulingproblemoftenreducestoaneconomiclotschedulingproblem(seeExample1.1.4).Thereareapplicationsoflotschedulingintheserviceindustriesaswell.Intheretailindustry(e.g.,Sears,Wal-Mart)theprocurementofeachitemhastobecontrolledcarefully.Placinganorderforadditionalsuppliesentailsanorderingcostandkeepingasupplyininventoryentailsaholdingcost.Theretailerhastodeterminethetrade-offbetweentheinventoryholdingcostsandtheorderingcosts.7.2OneTypeofItemandtheEconomicLotSizeInthissectionweconsiderthesimplestcase,namelyasinglemachineandonetypeofitem.Sincethereisonlyonetypeofitemthesubscriptjcan
7.2OneTypeofItemandtheEconomicLotSize145bedropped,i.e.,theproductionrateisQandthedemandrateisDitemsperunittime.Weassumethatthemachinecapacityissufficienttomeetthedemand,i.e.,Q>D.Theproblemistodeterminethelengthofaproductionrun.Afterarunhasbeenterminatedandsufficientinventoryhasbeenbuiltup,themachineremainsidleuntiltheinventoryhasbeendepletedandanewrunisabouttostart.Clearly,thelengthofaproductionrunisdeterminedbythetrade-offbetweeninventoryholdingcostsandsetupcosts.Inordertominimizethetotalcostperunittimewehavetofindanexpressionforthetotalcostoveracycle.Letxdenotethecycletimethathastobedetermined.IfDdenotesthedemandrate,thenthedemandoveracycleisDxandthelengthofaproduc-tionruntomeetthedemandoveracycleisDx/Q.Iftheinventorylevelatthebeginningoftheproductionruniszero,thentheinventorylevelgoesupduringtherunatarateQ−Duntilitreaches(Q−D)DxQ.DuringtheidleperiodtheinventorylevelgoesdownatarateDuntilitreacheszeroandthenextproductionrunstarts.Sotheaverageinventorylevelis12Dx−D2xQ.Eachproductionrunincursasetupcostc.Theaveragecostperunittimeduetosetupsisthereforec/x.Lethdenotetheinventoryholdingcostperitemperunittime.Thetotalaveragecostperunittimeduetoinventoryholdingcostsandsetupsistherefore12hDx−D2xQ+cx.Todeterminetheoptimalcyclelengthwetakethederivativeofthisexpressionwithrespecttoxandsetitequaltozero,yielding12hD1−DQ−cx2=0.Straightforwardalgebragivestheoptimalcyclelengthx=2QchD(Q−D).Thetotalamounttobeproducedduringacycle,i.e.,thelotsize,isDx=2DQch(Q−D).
1467EconomicLotSchedulingThelotsizeDxisnotnecessarilyanintegernumber.(Thisisoneofthedifferencesbetweencontinuousmodelsanddiscretemodels;thisdifferenceisexaminedmorecloselyinExample7.2.2.)Theidletimeofthemachineduringacycleturnsouttobex1−DQ.TheratioD/Qisattimesdenotedbyρ,andmayberegardedastheutilizationofthemachine,i.e.,theproportionoftimethatthemachineisbusy.NowconsiderthelimitingcasewhentheproductionrateQisarbitrarilyhigh,i.e.,Q→∞.Then,x=limQ→∞2QchD(Q−D)=2chD.InthiscasethelotsizeisequaltoDx=2Dch,whichisoftencalledtheEconomicLotSize(ELS)orEconomicOrderQuan-tity(EOQ).Alltheexpressionsabovearebasedontheassumptionthatthereisasetupcostbutnotasetuptime.If,inadditiontothesetupcost,thereisalsoasetuptimesands≤x(1−ρ),thenthesolutionpresentedaboveisstillfeasibleandoptimal.Ifs>x(1−ρ),thenthelotsizecomputedaboveisinfeasible.Theoptimalsolutionthenisthesolutionwherethemachinealternatesbetweensetupsandproductionrunswithacyclelengthx=s1−ρ.Thatis,themachineiseitherproducingorbeingsetupforthenextrun.Themachineisneveridle.Thefirstexampleillustratestheuseoftheseformulae.Example7.2.1(TheELSPwithandwithoutSetupTimes).ConsiderafacilitywithaproductionrateQ=90itemsperweek,ademandrateD=50itemsperweek,asetupcostc=$2000,aholdingcosth=$20peritemperweek,andnosetuptimes.Fromtheanalysisaboveitfollowsthatthecycletimexis3weeksandthequantityproducedinacycleis150.Figure7.1.adepictstheinventoryleveloverthecycle.Theidletimeduringacycleis3(1−5/9)=1.33weeks,whichisapproximately9days.Nowsupposethattherearesetuptimes.Ifthesetuptimeislessthan9days(thelengthoftheidleperiod),thenthe3weekcycleremainsoptimal.
7.2OneTypeofItemandtheEconomicLotSize14712345666.66123456100Time (weeks)Time (weeks)Quantityin stockQuantityin stock(a) No setup time(b) Setup time of 2 weeksFig.7.1.InventorylevelsinExample7.2.1Ifthesetuptimeislongerthan9days,thenthecycletimehastobelonger.Forexample,ifasetuplasts2weeks(becauseofmaintenanceandcleaning),thenthecycletimeis4.5weeks.Figure7.1.bdepictstheinventoryleveloveracycle.Thenextexamplehighlightsthedifferencesbetweencontinuousanddis-cretesettings.Example7.2.2(ContinuousSettingvs.DiscreteSetting).ConsideraproductionrateQof0.3333itemsperday,aholdingcosthof$5.00peritemperdayandasetupcostcof$90.00.ThedemandrateDis0.10itemsperday.Applyingthecyclelengthformulagivesx=600.5(0.3333−0.1)=22.678andthenumberofitemsinalotisDx=2.2678.Inadiscretesettingsuchanumberisnotfeasible.Considerthefollowingdiscretecounterpartofthisinstance.Thetimetoproduceoneitem(orjob)isp=1/Q=3days.Thedemandrateis1itemevery10days.Alotofsizek,kinteger,hastobeproducedevery10kdays.(Thesolutioninthecontinuous
1487EconomicLotSchedulingsettingsuggeststhattheoptimalsolutioninthediscretesettingiseitheralotofsize2every20daysoralotofsize3every30days.)Thetotalcostperdaywithalotofsize1every10daysis90/10=$9.00.Thetotalcostperdaywithalotofsize2every20daysis(90+7×5)/20=$6.25andthetotalcostperdaywithalotofsize3every30daysis(90+7×5+14×5)/30=$6.50.Soinadiscretesettingitisoptimaltoproduceevery20daysalotofsize2.7.3DifferentTypesofItems-RotationSchedulesConsideragainasinglemachine,butnowwithndifferentitems.ThedemandrateforitemjisDjandthemachineiscapableofproducingitemjatarateQj.Inordertostartaproductionrunforitemj,asetupcostcjisincurred.Weassume,forthetimebeing,thatthissetupcostissequenceindependent.Inthissectionwedeterminethebestproductioncyclethatcontainsasinglerunofeachitem.Thus,thecyclelengthsofthenitemshavetobeidentical.Suchascheduleisreferredtoasarotationschedule.Thelengthofthecycledeterminesthelengthofeachoftheproductionruns.Hencethereisonlyasingledecisionvariable,thecyclelengthx.Inordertodeterminetheoptimalcyclelength,itisagainnecessarytofindanexpressionforthetotalcostperunittimeasafunctionofthecyclelengthx.Ifsetupsrequiremachinetime,thenitmaynotbepossibletomakethecyclelengtharbitrarilysmallsincefrequentsetupsmaytakeuptoomuchmachinetime.ThelengthoftheproductionrunofitemjinacycleisDjx/Qj.Assumethattheinventorylevelatthebeginningoftheproductionrunofitemjiszero.Duringtheproductionrun,thelevelincreasesatrateQj−Djuntilitreacheslevel(Qj−Dj)Djx/Qj.Duringtheidleperiod,theinventorydecreasesatarateDjuntilitreacheszeroandthenextproductionrunstarts.Sotheaverageinventorylevelofitemjis12Djx−D2jxQj.Thefacilityincursasetupcostcjforeachproductionrunofitemj.Theaveragecostperunittimeduetosetupsforitemjisthereforecj/x.Thetotalaveragecostperunittimeduetoinventoryholdingcostsandsetupcostsisthereforenj=112hjDjx−D2jxQj+cjx.
7.3DifferentTypesofItems-RotationSchedules149Tofindtheoptimalcyclelengthwetakethederivativewithrespecttoxandsetitequaltozero,obtainingnj=112hjDj1−DjQj−nj=1cjx2=0,Straightforwardalgebrayieldstheoptimalcyclelengthx=nj=1hjDj(Qj−Dj)2Qj−1nj=1cj.Themachineidletimeduringacyclecanbecomputedinamannersimilartothesingleitemcase.Thisidletimeisequaltox1−nj=1DjQj.Theratioρj=Dj/Qjcanberegardedastheutilizationfactorofthemachineduetoitemj.Considerthelimitingcasewheretheproductionratesarearbitrarilyfast,i.e.,Qj=∞forj=1,…,n.Inthisspecialcasetheoptimalcyclelengthisx=nj=1hjDj2−1nj=1cj.Example7.3.1(RotationScheduleswithoutSetupTimes).Considerfourdifferentitemswiththefollowingproductionrates,demandrates,holdingcostsandsetupcosts.items1234Dj50506060Qj400400500400hj20203070cj200025008000Theoptimalcyclelengthxis1.24monthsandthetotalidletimeis0.48x=0.595months.Figure7.2displaystheoptimalrotationschedule.Thetotalaveragecostperunittimecanbecomputedeasilyandis2155+2559+1627+2213=8554.Asthesetupcostofitem4iszero,itisclearthatarotationscheduledoesnotmakesensehere.Itmakesmoresensetospreadtheproductionofitem4uniformlyoverthecycletoreduceinventoryholdingcosts.Inthenextsectionweconsiderthisexampleagainandallowformoregeneralschedules.
1507EconomicLotSchedulingTime (months)Idle timeInventorylevelsCycle timeItem 1Item 3Item 2Item 410080604020000.511.52Fig.7.2.RotationscheduleinExample7.3.1Intheanalysisabovetheorderinwhichthedifferentrunsaresequenceddoesnotmatter.Weassumedthattherewerenosetuptimesandthatsetupcostsweresequenceindependent.So,uptonow,therewasnotanyschedulingproblem,onlyalotsizingproblem.Iftherearesetuptimesthataresequenceindependent,i.e.,sjk=skforalljandk,thentheproblemstilldoesnothaveasequencingcomponent,sincethesumofthesetuptimesdoesnotdependonthesequence.Ifthesumofthesetuptimesislessthantheidletimeintherotationschedulecomputedabove,thelengthoftherotationscheduleremainsoptimal.Ifthesumofthesetuptimesexceedstheidletimecomputedabove,thentheactualoptimalcyclelengthhastobelargerthantheoptimalcyclelengthobtainedbefore.Actually,theoptimalcyclelengthagainturnsouttobethecyclelengththatcorrespondstoascheduleinwhichthemachineisneveridle,i.e.,x=nj=1sj1−nj=1ρj.Ifthesetuptimesaresequencedependent,thenthereisasequencingproblemandasequencethatminimizesthesumofthesetuptimeshastobefound.Minimizingthesumofthesetuptimesinarotationscheduleisequivalenttotheso-calledTravellingSalesmanProblem(TSP),whichcanbedescribedasfollows.Asalesmanhastovisitncitiesandthedistancefromcityjtocitykisdjk.Hisobjectiveistofindatourwiththeminimumtotaltraveldistance.ThatthisTSPisequivalenttooursequencingproblemcanbeshowneasily.Cityjcorrespondstoitemjandthedistancefromcityjto
7.3DifferentTypesofItems-RotationSchedules151cityk,djk,isequivalenttothesetuptimeneededwhenitemkfollowsitemj,i.e.,sjk.TheTSPisknowntobeNP-hard.If,inthecaseofsequencedependentsetuptimes,asequencecanbefoundthatminimizesthesumofthesetuptimesandthissumislessthantheidletimeintherotationschedule,thenthelotsizescomputedabove,aswellasthesequence,areoptimal.However,iftheoptimalsequenceresultsinatotalsetuptimethatislargerthanthemachineidletimeobtainedbefore,thentheoptimalcyclelengthhastobelargerthanthecyclelengthgivenintheformulaabove.Theoptimalcyclelengththenagainwillbesuchthatthemachineisalwayseitherproducingorbeingsetupforthenextproductionrun.Inanycase,thelotsizingproblemandtheschedulingproblemcanstillbeanalyzedseparately.Theschedulingproblemwitharbitrarysetuptimesisknowntobeex-tremelyhard.However,whenthesetuptimeshaveaspecialstructure,aneasysolutionmayexist.Consider,forexample,thefollowingsetuptimes:sjk=0,j≤kandsjk=(j−k)s,j>k.Anoptimalsequencecanbeobtainedbystartingoutwiththeitemwiththelowestindex,continuingwiththeitemwiththesecondlowestindex,andsoon.Attheendoftherunoftheitemwiththehighestindex,achangeoverismadetotheitemwiththelowestindexinordertostartanewcycle.ThissequenceisobtainedbyapplyingtheShortestSetupTimefirst(SST)rule,whichisoftenusedasaheuristicincaseswitharbitrarysetuptimes(seeAppendixC).Example7.3.2(RotationScheduleswithSetupTimes).ConsiderthesamefouritemsasinExample7.3.1.However,therearenowsequencedepen-dentsetuptimes.Thereare3!=6possiblesequences.Thesetuptimesaregiveninthetablebelow.k1234s1k-0.0640.4050.075s2k0.448-0.3190.529s3k0.0430.234-0.107s4k0.1450.1480.255-Thissetupmatrixisasymmetric,i.e.,sjkisnotnecessarilyequaltoskj.Recallthatinthecasewithoutsetuptimesthecycletimeis1.24monthsandthetotalidletimeis0.595months.Sincethereareonlysixsequences,allsequencescanbeenumeratedandthebestonecanbeselected.Thesequence1,4,2,3requiresatotalsetupof0.585months,whichisfeasibleandthereforeoptimal.However,ifSSTisusedstartingwithitem1,thenthesequence1,2,3,4isselected.Thissequencerequiresatotalsetupof0.635monthswhichexceedstheidletimeundertheoptimalcycle.
1527EconomicLotScheduling7.4DifferentTypesofItems-ArbitrarySchedulesWenowgeneralizethemodeldescribedintheprevioussectiontoallowforschedulesthataremoregeneralthanrotationschedules.Withinacycletheremaybemultiplerunsofanygivenitem.Forexample,iftherearethreediffer-entitems1,2and3,thenthecycle1,2,1,3isallowed.Theremaybesetupcostsaswellassetuptimes.IfρjdenotestheutilizationfactorDj/Qjofitemj,thenafeasiblesolutionexistsifandonlyifρ=nj=1ρj<1.ThisnecessaryandsufficientconditionisthesameastheconditionforthemodelinSection7.3.Itisintuitivethatthesetuptimesdonothaveanyimpactonthisfeasibilitycondition.Thesetuptimesdotakeupmachinetime;but,ifamachineisoperatingclosetocapacity,thenthecycletimeandindividualproductionrunsjusthavetobemadelongenoughinordertominimizetheimpactofthesetuptimes.Incontrasttothemodelintheprevioussectionforwhichthereexistsaclosedformsolution(atleast,whenthesetuptimesaresequenceindependent),theproblemconsideredinthissectionisveryhard.Theredoesnotexistanefficientalgorithmforthisproblem.However,therearegoodheuristicsthatusuallyleadtosatisfactorysolutions.Inwhatfollows,wedescribeonesuchheuristicforthecasewithsequenceindependentsetuptimes,i.e.,sjk=sk.LetSdenotethesetofallpossiblesequencesofarbitrarylengthandjltheindexoftheitemproducedinpositionlofthesequence.Soj1,...,jν,whereν≥n,denotestheproductionsequenceofagivencycle.Thesequencemaycontainrepetitions.Considertheitemthatisproducedinthel-thpositionofthesequence.Ifjl=k,thenitemkisproducedinthel-thpositionofthesequence.Intheremainderofthissection,thesuperscriptlisusedtorefertodatarelatedtotheitemproducedinthel-thpositionofthesequence,e.g.,Ql=Qjlandiftheiteminthel-thpositionisitemk,thenQl=Qjl=Qk.Theproductionoftheiteminpositionlinvolvesasetupcostcl,asetuptimesl,aproductiontimeτl,andasubsequentidletimeulwhichmaybezero.Ifitemkisproducedinthel-thposition,itemkmaybeproducedagainwithinthesamecycle.Letxdenotethecycletimeandvthetimefromthestartoftheproductionofitemkinthel-thpositiontillthestartofthenextproductionofitemk(thismaybeinthesamecycleorinthenextcycle).Sov=QlτlDlandifjl=k,thenv=QkτlDk. 7.4DifferentTypesofItems-ArbitrarySchedules153Thehighestinventorylevelis(Ql−Dl)τl.Thetotalinventorycostfortheproductionrunofitemkinpositionlis12hl(Ql−Dl)QlDl(τl)2.LetIkdenotethesetofallpositionsinthesequenceinwhichitemkispro-ducedandLlallthepositionsinthesequencestartingwithpositionl(whenitemkisproduced)upto,butnotincluding,thepositioninthesequencewhereitemkisproducednext.ThedefinitionofLlassumesthatthesequencej1,...,jνrepeatsitself.LetSdenotethesetofallpossiblecyclicschedules.TheELSPcannowbewrittenasminSminx,τl,ul1xνl=112hl(Ql−Dl)QlDl(τl)2+νl=1clsubjecttoj∈IkQkτj=Dkxfork=1,...,n,j∈Ll(τj+sj+uj)=QlDlτlforl=1,...,ν,νj=1(τj+sj+uj)=xThefirstsetofconstraintsensuresthatenoughtimeisallocatedtothepro-ductionofitemktomeetitsdemandoverthecycle.Thesecondsetensuresthatenoughoftheiteminpositionlisproducedtomeetthedemandtillthenexttimethatitemisproduced.Theproblemdescribedabovemaybeviewedasbeingcomposedofamas-terproblemandasubproblem.Themasterproblemfocusesonthesearchforthebestsequencej1,...,jν(anelementofS),andthesubproblemmustde-terminetheoptimalproductiontimes,idletimes,andcyclelength(τl,ul,x)giventhesequence.Thatthesubproblemisrelativelysimplecanbearguedasfollows.Ifthesequencej1,...,jνisfixed,thenthefirstsetofconstraintsinthenonlinearprogrammingformulationisredundant,sincesubstitutionofthethirdsetintothefirstsetyieldsj∈IkQjDjτj=νj=1(τj+sj+uj),whichisthesumofthesecondsetoverallpositionsinIk.So,givenafixedsequence,thenonlinearprogrammingproblemthatdeterminestheoptimalproductiontimesandidletimescanbeformulatedasfollows: 1547EconomicLotSchedulingminx,τl,ul1xνl=112hl(Ql−Dl)QlDl(τl)2+νl=1clsubjecttoj∈Ll(τj+sj+uj)=QlDlτlforl=1,...,ν,νj=1(τj+sj+uj)=xThemasterproblem,i.e.,findingthebestsequencej1,...,jν,ismorecomplicated.Oneparticularheuristicyieldsgoodsequencesinpractice.ThisheuristicisinwhatfollowsreferredtoastheFrequencyFixingandSequencing(FFS)heuristic.ThisFFSheuristicconsistsofthreephases:(i)Thecomputationofrelativefrequenciesphase.(ii)Theadjustmentofrelativefrequenciesphase.(iii)Thesequencingphase.Thefirstphasedeterminestherelativefrequencieswithwhichthevariousitemshavetobeproduced.Thenumberoftimesitemkisproducedduringacycleisdenotedbyyk.Inthesecondphase,theseproductionfrequenciesareadjustedsotheycanbespacedoutevenlyoverthecycle;theadjustedfrequencyofitemkisdenotedbyyk.Inthethirdandlastphasetheseadjustedfrequenciesareusedtoproduceanactualsequence.Thefirstphasedetermines,besidestherelativefrequenciesyk,alsothecorrespondingruntimesτk.Iftherunsofitemkareofequallengthandevenlyspaced,thenthefrequencyykandthecycletimexdeterminetheruntimeτk,i.e.,τk=ρkxyk.Tocomputetheykwerelaxtheoriginalnonlinearprogrammingformulationbydroppingthesecondsetofthethreesetsofconstraints.Withouttheseinterlinkingconstraintstheactualsequenceisnolongerimportant.Optimizingoversequencesnowbecomesoptimizingoverthecycletimexandruntimesτkor,equivalently,overproductionfrequenciesyk.SubstitutionsleadtothefollowingmodificationsintheobjectivefunctionoftheoriginalnonlinearprogrammingformulationoftheELSPproblem:1xνl=112hl(Ql−Dl)QlDl(τl)2+νl=1cl=1xnk=112ykhk(Qk−Dk)QkDk(τk)2+nk=1ykck 7.4DifferentTypesofItems-ArbitrarySchedules155=nk=112hk(Qk−Dk)QkDkρkτk+nk=1ckρkτk=nk=112hk(Qk−Dk)τk+nk=1ckρkτk=nk=1akτkρk+nk=1ckρkτk=nk=1akxyk+nk=1ckykx,whereak=12hk(Qk−Dk)ρk=12hk(1−ρk)Dk.Ofcourse,inanyfeasiblescheduletherelativefrequenciesykhavetobeintegers.Thisimpliesthattheruntimesτkcannotassumejustanyvalues,sincetheyaredeterminedbytherelativefrequencies.However,inordertomaketheproblemeasier,wedeletetheintegralityconstraintsontheykandthusrelaxtheconstraintsontheτkaswell(basicallydeletingthefirstsetofconstraintsintheoriginalnonlinearprogrammingformulation).Disregardingtheintegralityconstraintsontheykresultsinarelativelyeasynonlinearprogrammingproblem.minyk,xnk=1akxyk+nk=1ckykx,subjecttonk=1skykx≤1−ρ.Theconstraintinthisproblemisequivalenttothelastconstraintintheoriginalformulation.Ifthelefthandsideofthisconstraintisstrictlysmallerthantherighthandside,thenthesumofthesetuptimesislessthanthetimethemachineisnotproducing,implyingthereisstillsomeidletimeremaining.Beforepresentingasolutionforthissimplifiednonlinearprogrammingproblemsomeobservationshavetobemade.First,asolutionthatisfeasibleforthesimplifiednonlinearprogrammingproblemmaynotbefeasiblefortheoriginalnonlinearprogrammingproblem;thesolutiontothesimplifiedproblemmayrequiresometweaking.Second,itisclearthatthesimplifiednonlinearprogrammingproblemhasaninfinitenumberofoptimalsolutions.Ifthesolutionx∗,y∗1,...,y∗n,isoptimalandnk=1skykx<1−ρ 1567EconomicLotScheduling(i.e.,theinequalityisstrict),thenanysolutionkx∗,ky∗1,...,ky∗n,withkin-teger,isalsooptimal.Basically,multiplyingthefirstcyclicschedulebyanintegerkresultsinanidenticalcyclicschedule.Athirdobservationisthefol-lowing:iftheinequalityaboveisstrict,thenthesolutionx∗,y∗1,...,y∗nwouldalsobeoptimalifallsetuptimesarezero.However,ifintheoptimalsolutionnk=1skykx=1−ρ,thenthesetuptimesplayanimportantrole.Thefactthatthereisnoidletimeimpliesthattheoptimalsolutionrequiresrelativelyhighproductionfrequen-cieswithrelativelyshortruntimes(possiblyduetohighholdingcostsorlowsetupcosts).Thesehighfrequenciesrequirethatthemaximumproportionofmachinetime,i.e.,(1−ρ),isdedicatedtosetuptimes.Thenonlinearprogrammingproblemcanbedealtwithasfollows.Incorpo-ratingtheconstraintintheobjectivefunctionusingaLagrangeanmultiplierλ(λ≥0),resultsinanunconstrainednonlinearoptimizationproblemwithobjectivefunctionminyk,xnk=1akxyk+nk=1ckykx+λnk=1skykx−(1−ρ).Takingthepartialderivativeofthisfunctionwithrespecttoykandsettingitequaltozeroyieldsyk=xakck+λsk.Thecyclelengthxcanbeadjustedsothattheproductionfrequenciestakeappropriatevalues(forexample,onemaychoosethecyclelengthxsothatthesmallestfrequencyvalueisapproximatelyequalto1).Ifthereareidletimes,thenλissetequaltozero.Iftherearenoidletimes,thentheλhastosatisfytheequationnk=1skakck+λsk=1−ρ,sincenk=1skyk=(1−ρ)x.Thesolutionykisunlikelytobeinteger.Tofindanintegersolutionthatisclosetothevaluesobtainedforykmayrequiretheconstructionofalongsequencewithhighfrequencies.ThesecondphaseoftheFFSheuristicmakesadjustmentsinthefrequen-ciesyk.Ithasbeenshownintheliteraturethatitispossibletofindanewsetoffrequenciesykthatareintegersandpowersof2withthecostofthis 7.4DifferentTypesofItems-ArbitrarySchedules157newsolutionbeingwithin6%ofthecostoftheoriginalsolution.Ofcourse,theruntimesofitemkhavetochangethenaswell.Thenewruntimes,τk,canbecomputedbyassumingthatthetotalidletimeremainsthesameandtherunsofitemkareofequallengthandequallyspaced.ThethirdphaseoftheFFSheuristicgeneratestheactualsequence.Theheuristicusedherehasitsrootsintheheuristicusedforschedulingndifferentjobsonanumberofparallelmachinestominimizethemakespan.(RecallfromthesecondsectionofChapter5thatthemostpopularheuristicforthisproblemistheLPTrule).Letymax=max(y1,...,yn).Foreachitemk,thereareykjobswiththesameestimatedprocessingtimeτk(assumingthatthelotswillbeequallyspaced).Nowconsideraschedulingproblemwithymaxmachinesinparallelandykjobsoflengthτk,k=1,...,n,(implyingatotalofnk=1ykjobs).Thereisanadditionalrestrictioninthatitemkwithfrequencyykmusthavetheyklots(jobs)placedonmachinesthatareequallyspaced.Forexample,ifymax=6andyk=3,thentherearetwochoices:thethreejobsareassignedeithertomachines1,3and5ortomachines2,4and6.NowthefollowingvariationoftheLPTheuristiccanbeused:thepairs(yk,τk)arelistedindecreasingorderofyk.Pairswithidenticalfrequenciesykarelistedindecreasingorderoftheestimatedprocessingtimeτk.Thepairsaretakenfromthelistonebyonestartingatthetop.Whenthepair(yk,τk)istakenfromthelist,thecorrespondingykjobsoflengthτkareputonthemachines(satisfyingthespacingrestriction)sothatthemaximumofthetotalprocessingassignedsofartotheselectedykmachinesisminimized.Afterallpairsinthelisthavebeenassigned,theresultingsequencesontheymaxmachinesareconcatenated,i.e.,machine1,followedbymachine2,andsoon,toobtainasinglesequence.Thisideaisbasedonthefactthatafteralljobshavebeenscheduledandthetotalprocessingismoreorlessequallypartitionedoverallthemachines,theconcatenatedsequencewillmaintaintheequalspacingofthevariousrunsofanygivenitem.ThefollowingexampleillustratestheapplicationoftheFFSheuristictoaninstancewithoutsetuptimes.Example7.4.1(ApplicationoftheFFSHeuristicwithoutSetupTimes).ConsideragainthesituationdescribedinExample7.3.1.However,nowthescheduledoesnotnecessarilyhavetobearotationschedule.Therearesetupcostsbutnosetuptimes.Sinceitem4hasnosetupcostandafairlyhighholdingcost,itsproductionshouldbespreadoutasuniformlyaspossibleinbetweentheproductionoftheotherthreeitems.Inordertofindthefrequenciesyk,wehavetofirstsolvetheunconstrainedoptimizationproblem.Notethat(1−ρ)x=0.48x, 1587EconomicLotSchedulingyk=ρkxτk,andak=12hk(Qk−Dk)ρk.Thefollowingvaluescanbecomputedeasily.items1234Dj50506060Qj400400500400hj20203070cj200025008000ρj0.1250.1250.120.15aj437.5437.57921785Itimmediatelyfollowsthaty1=0.46xy2=0.42xy3=0.99xy4=∞Supposethatthecycletimexissetequalto2months.Thiscycletimecorrespondstothefollowingapproximatevaluesfory1,...,y4:y1=y2=1,y3=2andy4=16.Thechoiceofy4issomewhatarbitrarybutithastobemadehigh.Thehighery4,themoreuniformtheproductionofitem4canbemadeinthefinalsolution.Givenacycletimeof2monthsandtheproductionfrequenciesabove,theruntimesτkofthefouritemsareτ1=τ2=0.25,τ3=0.12,τ4=0.3/16.NowweapplytheLPT-likeheuristic.Thenumberofmachinesinparallelisymax=16.Item4,thefirsttobeassigned,isassignedtoall16machineswithall16processingtimesequalto0.3/16.Item3isassignednextandisassignedtomachines1and9withthetwoprocessingtimesequalto0.12.Item1isthenputonmachine5anditem2onmachine13.Concatenatingthesequencesofthe16parallelmachinesresultsinthecyclicschedule|4,3|4|4|4|4,1|4|4|4|4,3|4|4|4|4,2|4|4|4|.Item4goesfirstfor0.3/16months,followedbyitem3for0.12months.Item4goesnextfourtimesinarow,eachtimefor0.3/16months(thesefourrunsareseparatedinthefinalsolutionbyidletimes).Item1followsfor0.25months.Item4goesagainfourtimes,andsoon.Afeasibilitycheckhastobedone.Itisclearthat,inanidealsolution,therunsofeachitemarespacedevenlyoverthecycle.Iftherunsofanitemareevenlyspaced,weareassuredthattherearenostockouts.Attemptingto 7.4DifferentTypesofItems-ArbitrarySchedules15910080604020000.511.5213332Time (months)InventorylevelsItem 1Item 3Item 2Item 4Fig.7.3.ScheduleinExample7.4.1uniformizetheproductionofitem4overthecycleresultsinmanyshortrunsthatareeitherseparatedbyidletimesorbyproductionrunsofotheritems.Weneedtocheckwhether,wheneveritems1,2,or3areproduced,thestockofitem4issufficienttocoverthedemandintheperiodsthattheotheritemsareinproduction.Thissolutioncouldhavebeenobtainedinanotherwayaswell.Asmen-tionedabove,they4wasselectedsomewhatarbitrarily.Thereasonforchoos-ingahighvalueisthatitenablesustouniformizetheitem’sproductionoverthecycle.Itisclearthatthey4hastobechosenatleastaslargeasthesumofalltheothery’s,i.e.,atleast4.Ify4=4,thenthealgorithmyieldsthesequence4,3,4,1,4,3,4,2.Aschedulecannowbeconstructedasfollows.Firsttheproductionrunsofitems1,2,and3arespacedevenlyoverthecycleandfixed.Thentheproductionofitem4isscheduledseparatelyinbetweentheremainingidletimes.Thisproductionofitem4isscheduledevenlyovertime(inmanyshortruns).However,beforeanyoftheitems1,2,or3hastogointoproduc-tion,agivenamountofinventoryofitem4hastobebuiltup.Theinventorylevelofitem4dependsonthelengthoftherunsofitems1,2,and3.Itisonlyduringthesebuildupsofinventorythatholdingcostsareincurredforitem4.TheentirescheduleisdepictedinFigure7.3.Theaveragetotalcostperunittimecanbecomputedandisequalto1875+2125+1592+190=5782.RecallthatthecostoftherotationscheduleinExample7.3.1is8554. 1607EconomicLotSchedulingThenextexampleillustratestheapplicationoftheFFSheuristiconaninstancewithsetuptimes.Incontrasttothepreviousexamplethereisnow,undertheoptimalsequence,noidletimeonthemachine.Example7.4.2(ApplicationoftheFFSHeuristicwithSetupTimes).Considertheinstanceinthepreviousexamplebutnowwithsetuptimes.Thesetuptimesaresequenceindependent.items1234Dj50506060Qj400400500400hj20203070cj200025008000sj0.50.20.10.2ρj0.1250.1250.120.15aj437.5437.57921785Notethat,becauseofthenonzerosetuptimeofitem4thefrequencyofitem4cannotbemadearbitrarilyhigh.Inordertofindthefrequenciesyk,wefirsthavetofindaλthatsatisfiestheequationnk=1skakck+λsk=1−ρ.Itcanbeverifiedeasilythatλ≈8000satisfiesthisequation.Withthisvalueofλtheykfrequenciescanbecomputedasafunctionofthecycletimex,i.e.,yk=xakck+λsk.y1=0.27xy2=0.33xy3=0.70xy4=1.05xIfthecycletimexisfixedat3months,thentheapproximatevaluesy1,...,y4canbeeither(1,1,2,2)or(1,1,2,4).Bothsolutionsarepoweroftwosolutions.Comparethesesolutionswiththefrequencyvaluesinthepreviousexample,i.e.,(1,1,2,16).Itisclearthat,becauseofthesetuptimes,thefrequencyofitem4cannotbeashighasinthepreviousexample.Thereissimplynotenoughidletimeforthatmanysetups.Considerthesolutionx=3withfrequencies(1,1,2,2).Theitemsequenceis1,3,4,2,3,4.Ithastobeverifiedwhetherthissolutionisfeasible.Theidletimebeforetakingsetupsintoaccountis0.48×3=1.44monthsandthe 7.5MoreGeneralELSPModels161totalamountofsetuptimerequiredis1.3,whichimpliesthatthescheduleisfeasible.Actually,thismeansthatthecycletimexcanbemadeslightlysmallerthan3,andaslightlysmallercycletimewillgiveabettersolution.(Inthepreviousexample,withoutsetuptimes,thecyclelengthwas2months;herethecycletimewasmade3monthsassumingthattherewouldnotbeanyidletime.)Theaveragetotalcostperunittimecanbecomputedasinthepreviousexamples.Considerthesolutionx=3withfrequencies(1,1,2,4).Theorderoftheitemsis1,4,3,4,2,4,3,4.Thetotalsetuptimerequiredduringacycleis1.7months.Thisimpliesthatacyclelengthof3monthsisnotfeasiblewiththesefrequencies.Inordertohavethesefrequenciesthecyclelengthhastobemadelarger(seeExercise7.10).7.5MoreGeneralELSPModelsAllmodelsconsideredintheprevioussectionsaresinglemachinemodels.Someofthesemodelscanbeextendedfairlyeasily.Forexample,considerthemodelwithmultipleproductsonmidenticalmachinesinparallel.Therearesetupcostsbutnosetuptimes.Anyparticularitemhastobeprocessedononeandonlyoneofthemmachines.Theutilizationfactorofitemjisagaindefinedasρj=Dj/Qjandinorderforafeasiblesolutiontoexistwemusthavenj=1ρj≤m.Supposethatthescheduleforeachofthemachineshastobearotationschedule.Ifthecycletimesofthemrotationscheduleshavetobeequal,theproblemisrelativelyeasyandnotmuchdifferentfromtheonedescribedinSection7.3.Theonlyadditionalissuethatneedstoberesolvedistheassignmentoftheitemstothedifferentmachines.Assumingthateachitemhastobeproducedononeandonlyonemachine,theloadshavetobebalancedandthesumoftheρj’softheitemsassignedtoanyonemachinehastobelessthanone.Tofindagoodbalanceor,equivalently,agoodpartitionofthendifferentitemsoverthemmachines,wecanusetheLPTheuristicwiththeρjvaluesplayingtheroleofprocessingtimes.TheLPTheuristic(usedforminimizingthemakespaninaparallelmachineenvironment)willresultinareasonablygoodassignmentoftheitemstothemmachines.Example7.5.1(RotationScheduleswithMachinesinParallel).Con-siderthesituationinExample7.3.1.Insteadofasinglemachine,wenowhave2machinesinparallel.TheproductionrateofeachofthetwomachinesishalftheproductionrateofthemachineinExample7.3.1.Thedataarepresentedbelow. 1627EconomicLotSchedulingitems1234Dj50506060Qj200200250200hj20203070cj200025008000Becausethetwomachineshavethesamecyclelength,theformulainExample7.3.1canbeusedhereaswell.Theoptimalcyclexinthiscaseturnsouttobe1.35months(theoptimalcyclelengthwithasinglemachineattwicethespeedis1.24months).However,ifthecycletimesofthemrotationschedulesareallowedtobedifferent,thentheproblembecomesmoredifficult.Wecanreducetotalcostbytakingadvantageofdifferentcycletimes.Again,anassignmentoftheitemstothemachineshastobefoundwhilemaintainingapropermachineloadbalance.Thereisnowanadditionaldifficulty.Ifanitemwithacoststructurethatfavorsalongcycletimeisassignedtothesamemachineasanitemwithacoststructurethatfavorsashortcycletime,thenthesolutionisnotlikelytobeagoodone.Onecandealwiththisdifficultyasfollows.Considereachitemasasingleproductmodel(asinSection7.2)andcomputeitscycletime.Ranktheitemsindecreasingorderoftheircycletimes.Starttakingtheitemsfromthetopofthelistandputthemononemachine.Keepassigningitemstothismachineuntilitscapacityisexhausted,i.e.,theallocationofanitemmakesthesumoftheρj’slargerthanone.Thislastitemisthenreallocatedtothesecondmachine,andsoon.Thisproceduremaynotleadtoagoodloadbalance,andmayevenleadtoaninfeasiblesolution(i.e.,thesumoftheρj’sonthelastmachinemaybelargerthanone).Ifthatisthecase,thenitemsonadjacentmachineshavetobeswappedinordertoobtainabetterbalance.Theparallelmachinemodelwithrotationschedulesandsequencedepen-dentsetupsisofcourseharder.Theassignmentofitemstothemmachinesnowhastoconsidermachinebalance,preferredcycletimes,aswellassetuptimesonallmachines.Thesetuptimestructurebecomesespeciallyimportantwhentherearelargedifferencesbetweensetuptimes.Verylittleresearchhasbeendoneonthisproblem.Whentheschedulesonthedifferentmachinesdonothavetoberotationschedules,i.e.,theparallelmachinesgeneralizationofthemodelconsideredinSection7.4,theproblemisevenharder.However,nowitisnolongernecessarytoassignitemswithsimilarcycletimestothesamemachine.Itisclearthatintheparallelmachineenvironmenttherearestillmanyunresolvedissuesthatrequiremoreresearch.Anotherimportantextensionofthesinglemachinesettingistheenviron-mentwithmachinesinseries,i.e.,theflowshop.Considerasinglemachinefeedinganothersinglemachinewiththeproductionratesandsetupcostsofthetwomachinesbeingidentical.Hencethecoststructuresofanitemontheupstreammachineandonthedownstreammachinearethesame.Thema- 7.5MoreGeneralELSPModels163chinescanbescheduledandsynchronizedsothattheitems,whentheyleavetheupstreammachine,canimmediatelystarttheirprocessingonthedown-streammachinewithouthavingtowait.Becauseofthissynchronization,theresultsinSections7.3and7.4canbeextendedtothecaseofsimilarmachineswithidenticalcostsinseries.Example7.5.2(RotationScheduleswithMachinesinSeries).Con-siderthesameproductmixasinExample7.3.1.However,nowinsteadofasinglemachine,wehavetwomachinesinseries.Afteranitemhascompleteditsprocessingontheupstreammachine,ithastogotothedownstreamma-chineandcompleteitsprocessingthere.Therearesetupcostsbutnosetuptimes.Thesetupcostsofanitemonthetwomachinesarethesame.Arota-tionscheduleisneededforthetwomachines(i.e.,thesamerotationschedulemustbeusedforbothmachines).Iftherotationscheduleissuchthatanitemwithalongprocessingtime(i.e.,withalowproductionrate)isfollowedbyanitemwithashortpro-cessingtime(i.e.,withahighproductionrate),thentheitemwiththeshortprocessingtimemayhavetowaitbetweenthetwomachines.Assumethatatthebeginningoftherotationschedulemachine1startswiththeproductionoftheitemwiththeshortestprocessingtime(i.e.,withthehighestproduc-tionrate),itthencontinues,withoutstopping,withtheitemwiththesecondshortestprocessingtime,andsoon.Afteritcompletestherunoftheitemwiththelongestprocessingtimemachine1remainsidleuntilitisnecessarytostartthenewcycle.Inthiswayanyitemthatcomesoutofmachine1canstartimmediatelyonmachine2withouthavingtowait,i.e.,thereisnoWork-In-Processinbetweenthetwomachines.Theinventorycostsofthefinishedgoodsareexactlythesameasinthesinglemachinecase,sothissystemcanbeanalyzedasasinglemachine.However,therearenowtwosetupcosts,insteadofonlyone.Thisimpliesthattheoptimalcyclelengthis√2=1.4142timeslongerthantheoptimalcyclelengthforasinglemachine.Thisresultcanbeextendedeasilytomidenticalmachinesinseries.Theresultsinthepreviousexamplecanbefurthergeneralized.Considermmachinesinserieswithidenticalproductionratesforeachproducttype,butdifferentsetupcosts.Thisproblemcanstillbereducedtoasinglemachineproblemwithproductionratesidenticaltothoseofoneofthemachinesintheoriginalproblem.However,nowthesetupcostshavetobesetequaltothesumofthesetupcostsoftheoriginalmmachines,i.e.,thesetupcostforitemjinthenewproblemiscj=mi=1cij.Whenthemachinesdonothaveidenticalproductionratesforeachproducttypetheproblemisnotthateasy.Considerfirstthecasewithtwomachinesthathaveidenticalsetupcostsbutdifferentspeeds.Butthespeedstructure 1647EconomicLotSchedulingisuniformovertheitems,i.e.,theproductionrateofitemjonmachineiisQij=viQj,whereviisaspeedfactorofmachinei.Oneapproachistofirstanalyzetheslowmachineasasinglemachineinisolationandthenadaptthefastmachineaccordingly(sincethehighspeedmachineprovidesmoreflexibility).ThescheduleofthefastmachinecanbeadaptedinsuchawaythatthereisnoWIPinbetweenthetwomachines.Thismodelcanthenbeanalyzedasasinglemachinewiththeproductionratesoftheslowmachineandsetupcoststhatarethesumofthesetupcostsofthetwomachines.Whentheproductionratesarenotuniform,itmaybecomenecessarytoscheduleWork-In-Process(WIP)betweenthemachines.ThecarryingcostoftheWIPinbetweenthemachinesmaybedifferentfromtheholdingcostofthefinishedgoods.Thismakesthemodelmorecomplicated.Verylittleresearchhasbeendoneonthisproblem.Anevenmoregeneralmachineenvironmentistheflexibleflowshop,i.e.,anumberofstagesinserieswithateachstageanumberofmachinesinparallel.Underveryspecialconditionsoptimalrotationschedulescanbedeterminedforsuchamachineenvironment.Forexample,considertwostagesinserieswithateachstagetwomachinesinparallel.Foranygivenproducttypetheproductionratesofthefourmachinesarethesameandsoarethesetupcosts.UnderthesecircumstancesthereisnoneedforanyWIPinbetweentwostages.Thismakesitpossibletodeterminetheoptimalrotationschedulesrelativelyeasilywhenthecycletimesofallfourmachineshavetobethesame.7.6MultiproductPlanningandSchedulingatOwens-CorningFiberglasOwens-CorningFiberglasisaleadingmanufactureroffiberglassproductsandhasalargemanufacturingfacilityinAnderson,SouthCarolina.Initsman-ufacturingprocess,moltenfiberglassisformedandtheglassisspunontospoolsofvarioussizes.Thismaterialisusedtoweavefabricandtoproducechoppedstrandmat.Fiberglassmatissoldinrollsofvariouswidthsandweights,treatedwithoneofthreeprocessbinders,andtrimmedatoneorbothedgesornotatall.Thedemandfortheproductscomesmainlyfromthemarineindustryforthemanufactureofboathulls,andfromtheresidentialconstructionindustryforbathtubsandshowerbooths.Atthetimewhenaproductionplanningandschedulingsystemwasde-velopedforthisfacility,theproductlineconsistedofover200distinctmatitems.Twenty-eightoftheserepresentedover80%ofthetotalannualdemandandweretreatedashighvolumestandard(stock)products.Theremainingitemsweremadetoorder.Themanufacturingfacilityhadtwomainproduc-tionlinesreferredtoasMatLines1and2.Line1hadapproximatelythreetimesthecapacityofLine2andcouldproducemat76incheswide,whereasLine2waslimitedto60inches.Theproductcameofftheselinesintheformof175-to230-poundcylinders.TheaveragecostofdowntimeonLine1was 7.6MultiproductPlanningandSchedulingatOwens-CorningFiberglas165approximately$300perhour;theaveragecostofdowntimeonLine2wasless.Maintenancecostswererelatedtothefrequencyofjobchangeovers.Inaddition,eachtimeaproductchangewasmadeonaline,therewasasequencedependentsetupcostpartlyduetomaterialwaste.Themonthlycostsduetosetupsrangedfrom$15,000to$50,000(withapproximately75jobchangesand50hoursofdowntime).Theproductionplanningandschedulingsystemdevelopedforthematlinesfocusedonthreeissues,namelyaggregateplanning(focusingoninven-torycostsandworkforcescheduling),productionrunquantitiesandlotsizing(takinglineassignmentsandinventorylevelsintoaccount),anddetailedlinesequencingofMake-to-StockandMake-to-Orderproducts(takingsetupcostsintoaccount).Thesystemdevelopedforthematlinesconsistedthereforeofthreemainmodules,namely,(i)theaggregateplanningmodule,(ii)thelotsizingmodule,and(iii)thesequencingmodule.Theaggregateplanningmoduleusedasinputtheaggregatedemandforecastforthenexttwelvemonths.Itsobjectivewastominimizethesumofdirectpayrollcosts,overtimecostsandhiringandfiringcosts.Thetimehorizonrangedfromthreetotwelvemonths.Theoptimizationmethodinthismodulewasbasedonaproductionswitchingheuristic;thisruleconsideredinventorylevelsandforecastsoffuturedemandand,basedonthesedata,determinedtheappropriateproductionrates.Theoutputofthismodelincludedtargetswithrespecttoaggregateinventorylevels,productionratesandlevelsofem-ployment.Theoutputoftheaggregateplanningmoduleservedasinputtothelotsiz-ingmodule.Thelotsizingmodulealsorequiredadetailedshorttermdemandforecastforeachstockitem.Thetimehorizonconsideredinthismodulewasuptothreemonths.Theoutputfromthismodulewerethelineassignmentsandthelotsizes.Theoptimizationinthismodulewasbasedonalinearpro-gramformulation.Theobjectivewastominimizealltherelevantsetupcostsandproductioncostssubjecttoseveralsetsofconstraints.Theinventorylevelconstraintsensuredthataggregateinventorylevelsandsafetystocklevelsweremetandtheproductionbalanceconstraintsguaranteeddemandsatisfactionaswellasinventoryconservation.ThelinearprogramwassolvedusingtheMPSXpackageandtheprogramwasrunonamonthlybasisprovidingtheplantwithspecificinventorylevels,lotsizesandlineassignmentsforthecomingmonths.Theoutputofthelotsizingmoduleservedasinputforthesequencingmod-ule.Thetimehorizonofthismodulewasonemonth,anditsmainobjectivewastheminimizationofthesequencedependentsetupcosts.Thedominantcomponentsofsetupcostsweredirectdowntimeandmatwaste.Changeoverswereclassifiedasfiberchanges,widthchanges,weightchanges,andslitterchanges.Adistinctioncouldbemadewithineachfamilyofchangeoversbaseduponthedirectionofthechange.Forexample,itwaseasiertodecreaseweightandwidththantoincreasethem.Thesequencingheuristicwasbasedonsim- 1667EconomicLotSchedulingAggregateforecastsProductionswitchingheuristicLinearprogramLPIDailySchedulingMAt 1Forecasts forspecial ordersForecasts forstandardproductsAggregateinventorylevelsAggregateinventory levelsInventorycostsWorklevelWorkforcecostsPro-ductionratePro-ductioncostsPro-ductioncostsLot sizesProductionsequencesChangeovercostsOutputOutputOutputLot sizesLineassign-mentsLine assignmentsFig.7.4.OverviewofsystematOwens-CorningFiberglasspledispatchingrulesandthesequencingmodulewasrunonaweeklybasis.TheentiresystemisdepictedinFigure7.4.Implementationofthesystemledtomajorimprovementsintheoperationoftheplant.Theaveragenumberofchangeoverswentdownfromanaverageof70beforethesystem’simplementationtoanaverageof40afterthesystem’simplementation.TheOwens-CorningFiberglasenvironmentissomewhatsimilartotheset-tingdescribedinSection7.4.However,Owens-CorninghadtwomachinesinparallelinsteadofthesinglemachineinSection7.4(aparallelmachineenvi-ronmentwasconsideredinSection7.5).NotethattheFFSheuristicdescribedinSection7.4isbasedonanonlinearprogrammingformulation,whereasthelotsizingmoduleintheOwens-CorningFiberglassystemwasbasedonalinearprogrammingformulation.7.7DiscussionThemodelsdiscussedinthischapterhavesimilaritiesaswellasdifferenceswiththemodelsforthetheflexibleassemblysystemsdescribedinChapter6.Inbothchapterstheplanninghorizonsarebasicallyunbounded.(Thisisin 7.7Discussion167contrasttothemodelsdescribedinChapters4and5,whichallhaveafinitenumberofjobs.)However,theobjectivesconsideredinChapter6arefunda-mentallydifferentfromtheobjectivesconsideredinthischapter.InChapter6,thetypicalobjectiveistomaximizethethroughputor,equivalently,tominimizethecycletime.Inthischapter,thethroughputisbasicallygiven,sincethedemandlevelsareknown.Theobjectiveistominimizethesumoftheinventorycarryingcostsandthesetupcosts.Nonetheless,theobjectivesinthischapterdisplaysomesimilaritieswiththeobjectivesinpacedassemblysystems.ThemodelsconsideredinthischapterareveryimportantforindustriesthatproduceMake-To-Stockandforenvironmentswithsetuptimesandcosts.Examplesoftheseindustriesincludethepaperindustry,thealuminumindus-tryandthesteelindustry.Ifonecomparesthemodelsdescribedinthischapterwiththeproblemsthathavetobesolvedinthoseindustries,thenanumberofissuesarise.Theproblemsinpracticeare,ofcourse,morecomplicatedthanthemodelsconsideredinthischapter.Often,iftherearemultiplemachinesinparallel,theproductionratesofanygivenitemonthevariousmachinesmayvary.Runlengthshaveinpractice,besidesanimpactontheinventorycostsandthetotalchange-overcosts,alsoaneffectonvariousotherfactors,including(i)thequalityofthefinishedproduct,(ii)theproductionyieldortheamountofwasteincurred,(iii)theproductivityofthefacilityanditstotalproductioncapacity.Thesethreefactors,whicharenotindependent,areseldomincludedinmediumtermorlongtermplanningmodels.Thethreefactorsaresome-whatrelatedtooneanother.Thequalityoftheproductsinprocessindus-tries(whichtypicallyisacontinuousmeasureratherthanadiscretemea-sure)dependsstronglyonthelengthoftherun.Thelongerthelengthoftherun,thehighertheaveragequalityofproduction.Intheprocessin-dustriesthereisusuallyalsoayieldorwasteproblem(cuttingstockortrimrelatedissues).Iftherunlengthisverysmall,thentheaveragewastetendstoincrease.Sothemorechangeoversthereare,thelowertheav-eragequalityoftheproduct,andthelowertheproductivityandtheca-pacity.Thecostofmachinecapacityisbasicallydeterminedbyoppor-tunitycosts(i.e.,theshadowpricesordualpricesoftheresourcesin-volved).PracticalproblemsareoftenacombinationofMake-To-StockandMake-To-Order.TheMake-To-Stockaspectsinvolveproblemssuchasthosede-scribedinthischapterwhereastheMake-To-Orderaspectsarerelatedtojobshopschedulingproblems.Researchershavebeenanalyzinginventoryprob-lemsinwhichthefacilitiesareassumedtobesetupinseries.Thisareaofresearch,oftenreferredtoasmulti-echeloninventorytheoryorsupplychainmanagement,isconsideredinthenextchapter. 1687EconomicLotSchedulingExercises7.1.Considerfourdifferentproductswiththefollowingdemandrates,pro-ductionrates,holdingcostsandsetupcosts.items1234Dj50506060Qj400500500400hj20203070cj200010001000100(a)Findtheoptimalrotationschedule.Determineitscyclelength,andthetotalidletime.(b)Supposenowthatitem4canbeproducedmanytimesduringacycle.Items1,2and3stillcanbeproducedonlyonceduringacycle.Findtheoptimalproductionschedule.Howdoestheoptimalcyclelengthcomparewiththeoriginaloptimalcyclelength?7.2.Considertwoidenticalmachinesinparallel.Fouritemshavetobepro-duced.items1234Dj50506060Qj200200300300hj20203070cj200025008000(a)Findtheoptimalrotationscheduleassumingthatthecyclelengthsofthetwomachineshavetobethesame.Computethetotalaveragecostperunittime.(b)Findtheoptimalrotationschedulesofthetwomachinesassumingthecyclelengthsofthetwomachinesdonothavetobethesame(determinewhichitemshavetobecombinedwithoneanotheronthesamemachinetoobtainthebestresult).Computetheaveragecostperunittimeandcomparetheresultwiththeresultfoundunder(a).7.3.Considerthefollowingtwostageproductionprocessinapapermillwithadownstreamconvertingoperation.Atthefirststagethereisasinglepapermachine.Theoutputofthisoperationconsistsoflargerollsofpaper.Thesecondstageisasinglemachinecuttingoperationthatproducescutsizepaper.Tosimplifytheproblemassumethatonlytwoitemshavetobeproduced.Also,eachitemthatcomesoutofthesecondstagecorrespondstooneoftheitemsthatcomesoutofthefirststage.Theproductionrates,setupcostsandholdingcostsaredifferentatthetwostages.InthetablebelowQijdenotestheproductionrateofitemjatstagei,cijthesetupcostofitemjatstagei,andhijtheholdingcostofitemjafterprocessingatstagei(soh1jdenotes Exercises169theholdingcostofkeepingitemjininventoryinbetweenthetwostages,whileh2jdenotestheholdingcostofitemjasafinishedgood).items12Dj10050Q1j400400Q2j6001000h1j2020h2j6080c1j30002500c2j10001250Theschedulesatbothstageshavetoberotationschedules(i.e.,itis,forexample,notallowedtoproduceitem1atthefirststageforawhile,leavethemachineidleforsometime,produceitem1again,andthenitem2).(a)Assumingthatthecyclelengthxofthetwostageshavetobethesame,whatisthecyclelengthwiththeminimumtotalcost?(b)Assumethatthecyclelengthsatthetwostagesareallowedtobedifferent.Determinetheoptimalcyclelengthsofthetwostages.7.4.ConnsidertheenvironmentwithtwomachinesinparallelinExample7.5.1.Supposenowthattheproductionrateofonemachineis.7timestheproductionrateofthemachineinExample7.3.1andtheproductionrateofthesecondmachine.3times.a)Determinetheoptimalrotationscheduleswhenbothmachinesmusthavethesamecycletime.b)Determinetheoptimalrotationscheduleswhenthetwomachinesdonothavetohavethesamecycletimes.7.5.ConsiderthedatainExample7.3.1.Considernowmidenticalmachinesinparallel.TheproductionrateofitemjonanyoneofthemachinesisQj/m.(Thisimpliesthatthetotalproductioncapacitydoesnotdependonthenumberofmachinesinparallel.)Assumethatallthemachineshavetobescheduledaccordingtorotationscheduleswiththesamecycletimex.Computetheoptimalxandthetotalcostform=2,3,4.Plotthetotalcostagainstm.7.6.ConsidermidenticalfacilitieswitheachfacilityhavingaproductionrateQj/m.Therearenosetuptimes.Assumethatallthefacilitieshavetofollowrotationscheduleswiththesamecyclelengthx.Deriveanexpressionfortheoptimalcyclelengthx.Howdoesxdependonm?Discussthemonotonicityandtheconvexityofthefunction.7.7.Considerapapermillwithtwopapermachines.Thereare5differenttypesofpaperthathavetobeproduced.Items1and2havetobeproducedonmachine1anditem3hastobeproducedonmachine2.Items4and5canbeproducedoneitheroneofthetwomachines. 1707EconomicLotSchedulingitems12345Dj60608080100Qj200200300300400hj2030402020cj3000200080040001500(a)Determinetheoptimalrotationscheduleassumingthatthecyclelengthshavetobethesame.(b)Determinetheoptimalrotationschedulesassumingthecyclelengthsdonothavetobethesame.(c)Determinetheoptimalscheduleifthescheduleonmachine1hastobearotationscheduleandthescheduleonmachine2maybeanarbitraryschedule.7.8.ConsiderthefollowinggeneralizationofthepapermakingfacilityofEx-ercise7.3.Again,therearetwostages.Theentirefacilityproducesthreedif-ferentitems.However,thepapermachineatthefirststageproducesonlytwodifferentintermediateproducts.Oneoftheintermediateproductsthatcomesoutofstage1isusedatstage2toproduceitems1and2.(Thisimpliesthatthedatacorrepondingtoitems1and2regardingstage1arethesame.)Theotherintermediateproductthatcomesoutofstage1isconvertedatstage2intoitem3.items123Dj5070100Q1j250250300Q2j200400300h1j303040h2j402030c1j20002000900c2j100030002000Determinetheoptimalrotationscheduleanditscyclelength.7.9.ConsiderthesettinginExercise7.2.Insteadofasinglestagewithtwomachinesinparallel,wehavenowtwostagesinserieswithtwomachinesinparallelateachstage.Allfourmachinesareidenticalwithregardtopro-ductionratesandsetupcosts(thedatabeingthesameasinExercise7.2).Determinetheoptimalrotationschedulesofthefourmachinesassumingthatthecycletimesofthefourmachinesarethesame.7.10.ConsiderthesettinginExample7.4.2.(a)Whatistheminimumcycletimewhenthefrequencyvaluesarey1=y2=1andy3=y4=2?Computethetotalaveragecostofthissolution.(b)Whatistheminimumcycletimewhenthefrequencyvaluesarey1=y2=1,y3=2andy4=4?Computethetotalaveragecostofthissolution. CommentsandReferences171(c)Comparetheresultsobtainedunder(a)and(b)withthetotalaveragecostobtainedinExample7.4.1.Explainyourresults.CommentsandReferencesVariousbooksandmonographsfocusonlotsizingandscheduling;see,forexample,Haase(1994),Br¨uggemann(1995),Kimms(1997),andZipkin(2000).Severalexcel-lentsurveypaperscoverthistopicalsoindepth,seeGraves(1981)andDrexlandKimms(1997).ThematerialinSection2isverybasic.TheEOQformulawasfirstmentionedbyHarris(1915)andstudiedindetailbyWilson(1934).Thismaterialiscoveredineveryelementarytextbookonproductionplanningandoperationsmanagement.Maxwell(1964)didanexhaustivestudyofrotationschedules.Gallego(1988)andGallegoandRoundy(1992)generalizedtheseresultsallowingforbackordercosts.JonesandInman(1989,1996)madeanindepthstudyoftheworstcasebehaviorofrotationschedulesandcomparedrotationscheduleswithothertypesofschedules.GallegoandQueyranne(1995)extendedsomeoftheseresults.TheFFSheuristic,describedinSection7.4,forgeneratingarbitraryschedulesisduetoDobson(1987,1992).GallegoandShaw(1997)establishedtheNP-hardnessoftheELSPwitharbitrarycyclicschedules.Afairamountofworkhasbeendoneonlotschedulinginmorecomplicatedmachineenvironments;seeCrowston,WagnerandWilliams(1973),Carre˜no(1990),Br¨uggemann(1995),JonesandInman(1996),ChaoandPinedo(1996),andPinedoandChao(1999).TheproductionplanningandschedulingsystemdevelopedforOwens-CorningFiberglasisdiscussedinOliffandBurch(1985). Chapter8PlanningandSchedulinginSupplyChains8.1Introduction.................................1738.2SupplyChainSettingsandConfigurations......1758.3FrameworksforPlanningandSchedulinginSupplyChains................................1808.4AMediumTermPlanningModelforaSupplyChain........................................1868.5AShortTermSchedulingModelforaSupplyChain........................................1928.6CarlsbergDenmark:AnExampleofaSystemImplementation..............................1958.7Discussion...................................1998.1IntroductionThischapterfocusesonmodelsandsolutionapproachesforplanningandschedulinginsupplychains.Itdescribesseveralclassesofplanningandschedulingmodelsthatarecurrentlybeingusedinsystemsthatoptimizesupplychains.Italsodiscussesthearchitectureofthedecisionsupportsys-temsthathavebeenimplementedinindustryandtheproblemsthathavecomeupintheimplementationandintegrationofsystemsforsupplychains.Intheimplementationsconsideredthetotalcostinthesupplychainhastobeminimized,i.e.,thestagesinthesupplychaindonotcompetewithoneanotherinanyform,butcollaborateinordertominimizetotalcost.Thischapterbasicallyembedsmediumtermplanningmodels,suchasthelotsizingmodelsdescribedinChapter7,anddetailedschedulingmodels,suchasthejobshopschedulingmodelsdescribedinChapter5,intoasingleframework.Themodelsinthischapterarequitegeneral.Thereisanetworkofinterconnectedfacilitiesandthedemandsforthevariousend-productsmay© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_8,173 1748PlanningandSchedulinginSupplyChainsnotbestationary.Theplanningandschedulingmaybedoneatthesamepointintimebutwithdifferenthorizonsandwithdifferentlevelsofdetail.Amediumtermproductionplanningmodeltypicallyoptimizesseveralconsecutivestagesinasupplychain(i.e.,amulti-echelonmodel),witheachstagehavingoneormorefacilities.Suchamodelisdesignedtoallocatetheproductionofthedifferentproductstothevariousfacilitiesineachtimepe-riod,whiletakingintoaccountinventoryholdingcostsandtransportationcosts.Aplanningmodelmaymakeadistinctionbetweendifferentproductfamilies,butoftendoesnotmakeadistinctionbetweendifferentproductswithinafamily.Itmaydeterminetheoptimalrunlength(or,equivalently,batchsizeorlotsize)ofagivenproductfamilywhenadecisionhasbeenmadetoproducethatfamilyinagivenfacility.Iftherearemultiplefami-liesbeingproducedinthesamefacility,thentheremaybesetupcostsandsetuptimes.Theoptimalrunlengthofaproductfamilyisafunctionofthetrade-offbetweenthesetupcostand/orsetuptimeandtheinventorycarryingcost.Themainobjectivesinmediumtermplanninginvolveinventorycarryingcosts,transportationcosts,tardinesscostsandthemajorsetupcosts.How-ever,inamediumtermplanningmodelitistypicallynotcustomarytotakethesequencedependencyofsetuptimesandsetupcostsintoaccount.Thesequencedependencyofsetupsisdifficulttoincorporateinsuchanintegerprogrammingformulationanditcanincreasethecomplexityoftheformula-tionsignificantly.Ashorttermdetailedschedulingmodelistypicallyonlyconcernedwithasinglefacility,or,atmost,withasinglestage.Suchamodelusuallytakesmoredetailedinformationintoaccountthanaplanningmodel.Itistypicallyassumedthatthereareagivennumberofjobsandeachonehasitsownparameters(includingsequencedependentsetuptimesandsequencedepen-dentsetupcosts).Thejobshavetobescheduledinsuchawaythatoneormoreobjectivesareminimized,e.g.,thenumberofjobsthatareshippedlate,thetotalsetuptime,andsoon.TherelatedmodelshavebeendiscussedinChapters5and6.Clearly,planningmodelsdifferfromschedulingmodelsinanumberofways.First,planningmodelsoftencovermultiplestagesandoptimizeoveramediumtermhorizon,whereasschedulingmodelsareusuallydesignedforasinglestage(orfacility)andoptimizeoverashorttermhorizon.Second,planningmodelsusemoreaggregateinformation,whereasschedulingmod-elsusemoredetailedinformation.Third,theobjectivetobeminimizedinaplanningmodelistypicallyacostobjectiveandtheunitinwhichitismea-suredisamonetaryunit;theobjectivetobeminimizedinaschedulingmodelistypicallyafunctionofthecompletiontimesofthejobsandtheunitinwhichitismeasuredisoftenatimeunit.Nevertheless,eventhoughtherearefundamentaldifferencesbetweenthesetwotypesofmodels,theyoftenhavetobeincorporatedintoasingleframework,shareinformation,andinteractextensivelywithoneanother. 8.2SupplyChainSettingsandConfigurations175Planningandschedulingmodelsmayalsointeractwithothertypesofmodels,suchaslongtermstrategicmodels,facilitylocationmodels,demandmanagementmodels,andforecastingmodels;thesemodelsarenotdiscussedinthischapter.Theinteractionswiththeseothertypesofmodelstendtobelessintensiveandlessinteractive.Inthischapter,weassumethatthephysicalsettingofthesupplychainhasalreadybeenestablished;theconfigurationofthechainisgiven,andthenumberoffacilitiesateachstageisknown.Supplychainsinthevariousindustriesareoftennotverysimilarandmayactuallygiverisetodifferentissuesandproblems.Thischapterconsidersapplicationsofplanningandschedulingmodelsinsupplychainsinvariousindustrysectors.Adistinctioncanbemadebetweentwotypesofindustries,namelythecontinuousmanufacturingindustries(whichincludetheprocessindustries)andthediscretemanufacturingindustries(whichinclude,forex-ample,automotiveandconsumerelectronics).Eachoneofthesetwomaincategoriesissubdividedintoseveralsubcategories.Thiscategorizationisusedbecauseofthefactthattheplanningandschedulingproceduresinthetwomaincategoriestendtobedifferent.Wefocusontheframeworksinwhichtheplanningandschedulingmodelshavetobeembedded;wedescribethetypeofinformationthathastobetransferredbackandforthbetweenthemodulesandthekindofoptimizationthatisdonewithinthemodules.Thischapterisorganizedasfollows.Thesecondsectiondescribesandcat-egorizessomeofthetypicalindustrialsettings.Thethirdsectiondiscussestheoverallframeworksinwhichplanningmodelsandschedulingmodelshavetobeembedded.Thefourthsectiondescribesastandardmixedintegerprogram-mingformulationofaplanningmodelforasupplychain.Thefifthsectioncoversatypicalformulationofaschedulingprobleminafacilitywithinasup-plychain.ThesixthsectiondescribesanactualimplementationofaplanningandschedulingsoftwaresystemattheDanishbeerbrewerCarlsbergA/S.Thelastsectionpresentstheconclusionsanddiscussesthecurrenttrendsinthedesignanddevelopmentofdecisionsupportsystemsforsupplychains.8.2SupplyChainSettingsandConfigurationsThissectiongivesaconciseoverviewofthevarioustypesofsupplychains.Itdescribesthedifferencesinthecharacteristicsandparametersofthevariouscategories.Itfirstdescribesthevariousdifferentindustrygroupsandtheirsupplychaincharacteristicsandthendiscusseshowthedifferentplanningandschedulingmodelsanalyzedintheliteraturecanbeusedinthemanagementofthesechains.Onecanmakeadistinctionbetweentwotypesofmanufacturingindustries,namely:(I)Continuousmanufacturingindustries(e.g.,theprocessindustries),(II)Discretemanufacturingindustries(e.g.,cars,semiconductors).Thesetwoindustrysectorsarenotallencompassing;theborderlinesaresome-whatblurryandmayoverlap.However,planningandschedulingincontinuous 1768PlanningandSchedulinginSupplyChainsmanufacturing(theprocessindustries)oftenhavetodealwithissuesthatarequitedifferentfromthoseindiscretemanufacturing.ContinuousManufacturing:Continuousmanufacturing(process)indus-triesoftenhavevarioustypesofoperations.Themostcommontypesofoper-ationscanbecategorizedasfollows:(I-a)Mainprocessingoperations(I-b)Finishingorconvertingoperations,MainProcessingOperationsinContinuousManufacturing(I-a):Themainproductionfacilitiesintheprocessindustriesare,forexample,papermills,steelmills,aluminummills,chemicalplants,andrefineries.Inpaper,steel,andaluminummillsthemachinestakeintherawmaterial(e.g.,wood,ironore,alumina)andproducerollsofpaper,steeloraluminum,whichafterwardsarehandledandtransportedwithspecializedmaterialhandlingequipment.Machinesthatdothemainprocessingoperationstypicallyhaveveryhighstartupandshutdowncostsandusuallyworkaroundtheclock.Amachineintheprocessindustriesalsoincursahighchangeovercostwhenithastoswitchoverfromoneproducttoanother.Variousmethodologiescanbeusedforanalyzingandsolvingthemodelsforsuchoperations,in-cludingcyclicschedulingproceduresandMixedIntegerProgrammingap-proaches.FinishingOperationsinContinuousManufacturing(I-b):Manyprocessindustrieshavesomeformoffinishingoperationsthatdosomeconvertingoftheoutputofthemainproductionfacilities.Thisconvertingusuallyinvolvescuttingofthematerial,bending,foldingandpossiblypaintingorprinting.Theseoperationsoften(butnotalways)producecommoditytypeitems,forwhichtheproducerhasmanyclients.Forexample,afinishingoperationinthepaperindustrymayproducecutsizepaperfromtherollsthataresuppliedbyapapermill.ThepaperfinishingbusinessisoftenamixtureofMake-To-Stock(MTS)andMake-To-Order(MTO).IfitoperatesaccordingtoMTO,thentheschedulingisbasedoncustomerduedatesandsequencedependentsetuptimes.Thisleadsoftentosinglemachineandparallelmachineschedulingmodels.IfitoperatesaccordingtoMTS,thenitmayfollowaso-calleds-SorQ-Rinventorycontrolpolicy.IfitisamixtureofMTOandMTS,thentheschedulingpoliciesbecomeamixtureofinventorycontrolanddetailedschedulingrules.DiscreteManufacturing:Thediscretemanufacturingindustrysectorisquitediverseandincludestheautomotiveindustry,theappliancesindustry,andthePCindustry.Fromtheperspectiveofplanningandschedulingadis-tinctioncanbemadebetweenthreedifferenttypesofoperationsinthissector.Thereasonformakingsuchadistinctionisbasedonthefactthatplanningandschedulinginthesethreesegmentsarequitedifferent.(II-a)Primaryconvertingoperations(e.g.,cuttingandshapingofsheetmetal), 8.2SupplyChainSettingsandConfigurations177(II-b)Mainproductionoperations(e.g.,productionofengines,PCBs,wa-fers),and(II-c)Assemblyoperations(e.g.,cars,PCs).PrimaryConvertingOperationsinDiscreteManufacturing(II-a):Pri-maryconvertingoperationsaresomewhatsimilartothefinishingoperationsintheprocessindustries.Theseoperationsmaytypicallyincludestamping,cutting,andbending.Theoutputofsuchanoperationisoftenaparticularpartthatiscutandbentintoagivenshape.Thereareusuallyfewoperationsdoneonsuchanitemandtheroutinginsuchafacilityisrelativelysimple.Thefinalproductcomingoutofaprimaryconvertingfacilityisusuallynotafinishedgood,butratherapartorapiecemadeofasinglematerial(boxes,containers,frames,bodypartsofcars,andsoon).ExamplesofthetypesofoperationsinthiscategoryarestampingfacilitiesthatproducebodypartsforcarsandfacilitiesthatproduceepoxyboardsofvarioussizesfortheplantsthatproducePrintedCircuitBoards.TheplanningandschedulingproceduresunderII-amaybesimilartothoseunderI-b.However,theymaybeheremoreintegratedwiththeoperationsdownstream.MainProductionOperationsinDiscreteManufacturing(II-b):Themainproductionoperationsarethoseoperationsthatrequiremultipledifferentop-erationsbydifferentmachinetoolsandtheproduct(aswellasitsparts)mayhavetofollowacertainroutethroughthefacilitygoingthroughvariousworkcenters.Capitalinvestmentshavetobemadeinvarioustypesofmachinetools(lathes,mills,chipfabricationequipment).Forexample,inthesemiconductorindustrywaferstypicallyhavetoundergohundredsofsteps.Theseoperationsincludeoxidation,deposition,andmetallization,lithography,etching,ionim-plantation,photoresiststripping,andinspectionandmeasurements.Itisof-tenthecasethatcertainoperationshavetobeperformedrepeatedlyandthatsomeordershavetovisitcertainworkcentersinthefacilityseveraltimes,i.e.,theyhavetorecirculatethroughthefacility.InsemiconductorandPrintedCircuitBoardmanufacturingtheoperationsareoftenorganizedinajobshopfashion.Eachorderhasitsownroutethroughthesystem,itsownquantity(andprocessingtimes)anditsowncommittedshippingdate.Anordertyp-icallyrepresentsabatchofidenticalitemsthatrequiressequencedependentsetuptimesatmanyoperations.AssemblyOperationsinDiscreteManufacturing(II-c:)Themainpur-poseofanassemblyfacilityistoputdifferentpartstogether.Anassemblyfacilitytypicallydoesnotaltertheshapeorformofanyoneoftheindividualparts(withthepossibleexceptionofthepaintingofparts).Assemblyop-erationsusuallydonotrequiremajorinvestmentsinmachinetools,butdorequireinvestmentsinmaterialhandlingsystems(andpossiblyroboticas-semblyequipment).Anassemblyoperationmaybeorganizedinworkcells,inassemblylines,oraccordingtoamixtureofworkcellsandassemblylines.Forexample,PCsareassembledinworkcells,whereascarsandTVsaretyp-icallyputtogetherinassemblylines.Workcellstypicallydonotrequireany 1788PlanningandSchedulinginSupplyChainssequencing,buttheymaybesubjecttolearningcurves.Inassemblyopera-tionsthataresetupinaline,thesequencingisbasedongroupingandspacingheuristicscombinedwithcommittedshippingdates.Theschedulesthataregeneratedbythegroupingandspacingheuristicstypicallyaffectnotonlythethroughputoftheline,butalsothequalityoftheitemsproduced.Supplychainsinbothcontinuousanddiscretemanufacturingmayhave,inadditiontothestagesdescribedabove,additionalstages.InasupplychaininaprocessindustrytheremaybeastageprecedingStageI-ainwhichtherawmaterialisbeinggatheredatitspointoforigination(whichmaybeaforestoramine)andtakentothemainprocessingoperations.TheremayalsobeadistributionstagefollowingstageI-b.Acompanymayhaveitsowndistributioncentersindifferentgeographicallocations,whereitkeepscertainSKUsinstockforimmediatedelivery.Thecompanymayalsoshipdirectlyfromitsmanufacturingoperationstocustomers.Asupplychaininadiscretemanufacturingindustryalsomayhaveothertypesofstages.TheremaybeastageprecedingthefirststageII-ainwhichrawmaterialisbeingcollectedatasupplier(whichmaybeanoperationofthetypeI-b)andbroughttoaprimaryconvertingoperation.TheremayalsobeastagefollowingstageII-cthatwouldconsistofdistributionoperations(e.g.,dealerships).Supplychainsinbothcontinuousanddiscretemanufacturingmayhaveseveralfacilitiesateachoneofthestages,eachonefeedingintoseveralfacilitiesatstagesdownstream.Theconfigurationofanentirechainmaybequitecomplicated:Forexample,theremaybeassemblyoperationsthatproducesubassembliesthathavetobefedintoamainassemblyoperation.Therearesomebasicdifferencesbetweentheparametersandoperatingcharacteristicsofthefacilitiesinthetwomaincategoriesdescribedabove.Severalofthesedifferenceshaveanimpactontheplanningandschedulingprocesses,includingthedifferencesin(i)theplanninghorizon,(ii)theclock-speed,(iii)thelevelofproductdifferentiation.(i)Theplanninghorizonincontinuousmanufacturingfacilitiestendstobelongerthantheplanninghorizonindiscretemanufacturingfacilities.Incontinuousaswellasindiscretemanufacturingtheplanninghorizonstendtobeshorterthemoredownstreaminthesupplychain.(ii)Theso-called”clock-speed”tendstobehigherinadiscretemanufac-turingfacilitythaninacontinuousmanufacturingfacility.Ahighclock-speedimpliesthatexistingplansandschedulesoftenhavetobechangedoradjusted;thatis,planningandschedulingismorereactive.Incontinuousaswellasindiscretemanufacturingtheclockspeedincreasesthemoredownstreaminthesupplychain.(iii)Indiscretemanufacturingtheremaybeasignificantamountofmasscustomizationandproductdifferentiation.Incontinuousmanufacturingmass-customizationdoesnotplayaveryimportantrole.ThenumberofSKUsindiscretemanufacturingtendstobesignificantlylargerthanthenumberofSKUsincontinuousmanufacturing.ThenumberofSKUstendstoincreasemoredownstreaminthesupplychain. 8.2SupplyChainSettingsandConfigurations179Theseoperatingcharacteristicsaresummarizedintable8.1.SectorProcessesTimeHorizonClock-SpeedProductDifferentiation(I-a)planninglong-mediumlowverylow(I-b)planning/schedulingmedium/shortmedium/highmedium/low(II-a)planning/schedulingmedium/shortmediumverylow(II-b)planning/schedulingmedium/shortmediummedium/low(II-c)schedulingshorthighhighTable8.1.OperatingcharcateristicsBecauseofthesedifferences,theplanningandschedulingissuesineachoneofthesectorstendtobequitedifferent.Table8.2presentsasummaryofthemodeltypesthatcanbeusedinthedifferentcategoriesaswellasthecorrespondingsolutiontechniques.SectorModelsSolutionTechniques(I-a)Lotsizingmodels(multi-stage);MixedIntegerProgrammingcyclicschedulingmodelsformulations(I-b)Singlemachineschedulingmodels;Batchscheduling;parallelmachineschedulingmodelsmixturesofinventorycontrolrulesanddispatchingrules(II-a)Singlemachineschedulingmodels;BatchschedulingandParallelmachineschedulingmodelsdispatchingrules(II-b)FlowShopandJobShopSchedulingIntegerProgrammingModelswithspecificroutingpatternsformulations;shiftingbottleneckheuristics;dispatchingrules(II-c)AssemblyLineModels;GroupingandSpacingWorkcellModelsHeuristics;Make-to-Order/Just-In-TimeTable8.2.Modeltypesandthecorrespondingsolutiontechniques.NotethatproblemsthathavecontinuousvariablesmayresultinMixedIntegerProgramming(MIP)formulations,whereasproblemsthathaveonlydiscretevariablesmayresultinpureIntegerProgramming(IP)formulations 1808PlanningandSchedulinginSupplyChains(orDisjunctiveProgrammingformulations).However,adiscreteprobleminwhichcertainvariablesassumelargevalues(i.e.,thenumberofunitstobeproduced)maybereplacedbyacontinuousproblem,resultinginaMixedIntegerProgrammingformulationratherthanapureIntegerProgrammingformulation.PlanningmodelstypicallyresultinMixedIntegerProgrammingformulationswithamixofcontinuousanddiscretevariables.Schedulingmod-elsusuallydonothaveanycontinuousvariables(theymayhavecontinuousvariableswhenpreemptionsandjobsplittingareallowed).Whentherearefewdiscretevariables,itmakesalotofsensetosolvetheLinearProgram-mingrelaxationoftheMIP.ThesolutionmayprovideausefullowerboundandmaygivesomeinsightsintothestructureoftheoptimalsolutionstotheMIP.IftheformulationoftheproblemisapureIntegerProgram(whichisoftenthecasewithaschedulingproblem),thensolvingthelinearrelaxationtypicallydoesnotprovidemuchofabenefit.8.3FrameworksforPlanningandSchedulinginSupplyChainsThemainobjectiveinasupplychainorproductiondistributionnetworkistoproduceanddeliverfinishedproductstoendconsumersinthemostcosteffectiveandtimelymanner.Ofcourse,thisoverallobjectiveforceseachoneoftheindividualstagestoformulateitsownobjectives.Sinceplanningandschedulinginaglobalsupplychainrequirestheco-ordinationofoperationsinallstagesofthechain,themodelsandsolutiontechniquesdescribedintheprevioussectionhavetobeintegratedwithinoneframework.Differentmodelsthatrepresentsuccessivestageshavetoex-changeinformationandinteractwithoneanotherinvariousways.Acontin-uousmodelforonestagemayhavetointeractwithadiscretemodelforthenextstage.Planningandschedulingproceduresinsupplychainsaretypicallyusedinseveralphases:afirstphaseinvolvesamulti-stagemediumtermplanningprocess(usingaggregatedata)andasubsequentphaseperformsashorttermdetailedschedulingateachoneofthosestagesseparately.Typically,wheneveraplanningprocedurehasbeenappliedandtheresultshavebecomeavailable,eachfacilitycanapplyitsschedulingprocedures.However,schedulingpro-ceduresareusuallyappliedmorefrequentlythanplanningprocedures.Eachfacilityineveryoneofthesestageshasitsowndetailedschedulingissuestodealwith,seeFigure8.1.Ifsuccessivestagesinasupplychainbelongtothesamecompany,thenitisusuallythecasethatthesestagesareincorporatedintoasingleplanningmodel.Themediumtermplanningprocessattemptstominimizethetotalcostoverallthestages.Thecoststhathavetobeminimizedinthisoptimiza-tionprocessincludeproductioncosts,holdingorstoragecosts,transportationcosts,tardinesscosts,non-deliverycosts,handlingcosts,costsforincreases 8.3FrameworksforPlanningandSchedulinginSupplyChains181Buffer Inventory may Decouple Medium Term PlanningMediumTermPlanningShortTermSchedulingUpstream DownstreamStage 1Stage 2Stage 3Stage 4Stage 5FacilityPlanning systemScheduling systemFig.8.1.PlanningandSchedulinginSupplyChainsinresourcecapacities(e.g.,schedulingthirdshifts),andcostsforincreasesinstoragecapacities.Inthismediumtermoptimizationprocessmanyinputdataareonlycon-sideredinanaggregateform.Forexample,timeisoftenmeasuredinweeksormonthsratherthandays.Distinctionsareusuallyonlymadebetweenma-jorproductfamilies,andnodistinctionsaremadebetweendifferentproductswithinonefamily.Asetupcostmaybetakenintoaccount,butitmayonlybeconsideredasafunctionoftheproductitselfandnotasafunctionofthesequence.Theresultsofthisoptimizationprocessaredailyorweeklyproductionquantitiesforallproductfamiliesateachlocationorfacilityaswellastheamountsscheduledfortransporteveryweekbetweenthelocations.Thepro-ductionoftheordersrequireacertainamountofthecapacitiesoftheresourcesatthevariousfacilities,butnodetailedschedulingtakesplaceinthemediumtermoptimization.Theoutputconsistsoftheallocationsofresourcestothevariousproductfamilies,theassignmentofproductstothevariousfacilitiesineachtimeperiod,andtheinventorylevelsofthefinishedgoodsatthevar-iouslocations.Asstatedbefore,inthisphaseoftheoptimizationprocessa 1828PlanningandSchedulinginSupplyChainsdistinctionmaybemadebetweendifferentproductfamilies,butnotbetweendifferentproductswithinthesamefamily.ThemodelistypicallyformulatedasaMixedIntegerProgram.Variablesthatrepresentproductionquantitiesareoftencontinuous.Integer(discrete)variablesareoften0-1variables;theyare,forexample,neededintheformulationwhenadecisionhastobemadewhetherornotaparticularproductfamilywillbeproducedatacertainfacilityduringagiventimeperiod.Theoutputofthemediumtermplanningprocessservesasaninputtothedetailed(shortterm)schedulingprocess.Thedetailedschedulingproblemstypicallyattempttooptimizeeachstageandeachfacilityseparately.Sointheschedulingphaseoftheoptimizationprocess,theprocessispartitionedaccordingto(i)thedifferentstagesandfacilitiesand(ii)thedifferenttimeperiods.Soineachdetailedschedulingproblemthescopeisconsiderablynarrower(withregardtotimeaswellasspace),butthelevelofdetailtakenintoconsiderationisconsiderablyhigher,seeFigure8.2.Thislevelofdetailisincreasedinthefollowingdimensions:(i)thetimeismeasuredinasmallerunit(e.g.,daysorhours);theprocessmaybeeventimecontinuous,(ii)thehorizonisshorter,(iii)theproductdemandismorepreciselydefined,and(iv)thefacilityisnotasingleentity,butacollectionofresourcesormachines.Theproductdemandnowdoesnotconsist,asinthemediumtermplanningprocess,ofaggregatedemandsforentireproductfamilies.Inthedetailedschedulingprocessthedemandforeachindividualproductwithinafamilyistakenintoaccount.Theminorsetuptimesandsetupcostsinbetweendifferentproductsfromthesamefamilyaretakenintoaccountaswellasthesequencedependency.Thefacilityisnownotasingleentity;eachproducthastoundergoanum-berofoperationsondifferentmachines.Eachproducthasagivenrouteandgivenprocessingrequirementsonthevariousmachines.Thedetailedschedul-ingproblemcanbeanalyzedasajobshopproblemandvarioustechniquescanbeused,including(i)dispatchingrules,(ii)shiftingbottlenecktechniques,(iii)localsearchprocedures(e.g.,geneticalgorithms),or(iv)integerprogrammingtechniques.Theobjectivetakesintoaccounttheindividualduedatesoftheorders,sequencedependentsetuptimes,sequencedependentsetupcosts,leadtimes,aswellasthecostsoftheresources.However,iftwosuccessivefacilities(orstages)aretightlycoupledwithoneanother(i.e.,thetwofacilitiesoperateaccordingtotheJITprinciple),thentheshorttermschedulingprocessmayoptimizethetwofacilitiesjointly.Itactuallymayconsiderthemasasin- 8.3FrameworksforPlanningandSchedulinginSupplyChains183Transportation costsInventory costsTardiness costsEstimates of production data(processing time; due dates)(Medium Term)Tactical: -PlanningMultiple Stages of FacilitiesInventory costsTardiness costsSequence dependent setup costsExact production data(processing times; due dates)(Short Term)Operational SchedulingSingle Stage or FacilityDATA AGGREGATIONCONSTRAINT PROPAGATIONFig.8.2.Dataaggregationandconstraintpropogationglefacilitywiththetransportationinbetweenthetwofacilitiesasanotheroperation.Theinteractionbetweenaplanningmoduleandaschedulingmodulemaybeintricate.Aschedulingmodulemaycoveronlyarelativelyshorttimehori-zon(e.g.,onemonth),whereastheplanningmodulemaycoveralongertimehorizon(e.g.,sixmonths).Aftertheschedulehasbeenfixedforthefirstmonth(fixingthescheduleforthismonthrequiressomeinputfromtheplanningmodule),theplanningmoduledoesnotconsiderthisfirstmonthanymore;itassumesthescheduleforthefirstmonthtobefixed.However,theplanningmodulestilltriestooptimizetheseconduptothesixthmonth.Doingso,itconsiderstheoutputoftheschedulingmoduleasaboundarycondition(seeFigure8.3).However,italsomaybethecasethatthetimeperiodscoveredbythedetailedschedulingprocessandthemediumtermplanningprocessoverlap.Aplanningandschedulingframeworkforasupplychaintypicallymusthaveamechanismthatallowsfeedbackfromaschedulingmoduletotheplan-ningmodule,seeFigure8.4.Thisfeedbackmechanismenablestheoptimiza-tionprocesstogothroughseveraliterations.Itmaybeusedundervarious 1848PlanningandSchedulinginSupplyChainsMedium Term PlanningConsiders detailed scheduling as fixed.Capacity reduction because of fixed detailed schedule.Detailed SchedulingConsiders medium term planned demands as due dates.No capacity reduction from medium term planning orders.Detailed Scheduling HorizonMedium Term Planning HorizonFig.8.3.Schedulingandplanninghorizonscircumstances:First,theresultsofthedetailedshorttermoptimizationpro-cessmayindicatethattheestimatesusedasinputdataforthemediumtermplanningprocesswerenotaccurate.(Theaverageproductiontimesintheplanningprocessesdonottakethesequencedependencyofthesetuptimesintoaccount;setuptimesareestimatedandembeddedinthetotalproductiontimes.Thetotalsetuptimesinthedetailedschedulemayactuallybehigherthanthesetuptimesanticipatedintheplanningprocedure).Iftheresultsofthedetailedschedulingprocessindicatethattheinputtotheplanningprocesshastobemodified,thennewinputdatafortheplanningprocesshavetobegeneratedandtheplanningprocesshastoberedone.Second,theremaybeanexogenousreasonnecessitatingafeedbackfromthedetailedschedulingprocesstothemediumtermplanningprocess.Amajordisruptionmayoccurontheshopfloorlevel,e.g.,animportantmachinegoesdownforanextendedperiodoftime.Adisruptionmaybeofsuchmagnitudethatitnotonlyaffectsthefacilitywhereitoccurs,butotherfacilitiesaswell.Theentireplanningprocessmaybeaffected.Aframeworkwithafeedbackmechanismallowstheoveralloptimizationprocesstoiterate,seeFigure8.4.Theindividualmoduleswithintheplanningandschedulingframeworkforagivenchainmayhaveotherinterestingfeatures.Twotypesoffeaturesthatareoftenincorporatedaredecompositionfeaturesandso-calleddiscretizationfeatures.Eachfeaturecanbeactivatedanddeactivatedbytheuserofthesystem. 8.3FrameworksforPlanningandSchedulinginSupplyChains185ProductionData(estimates)Medium TermPlanning(multiple stages)Short termDetailed Scheduling(stage k)Short termDetailed Scheduling(stage l)Are Detailed SchedulesConsistent with Production Plans?yesImplementnoReviseProductionData(machine availabilities. processing times, etc.)Fig.8.4.InformationflowsbetweenplanningandschedulingsystemsDecompositionisoftenusedwhentheoptimizationproblemissimplytoolargetobedealtwitheffectivelybytheroutinesavailable.Adecompositionprocesspartitionstheoverallprobleminanumberofsubproblemsandsolvesthe(smaller)subproblemsseparately.Attheendoftheprocessthepartialsolutionsarestitchedtogetherintooneoverallsolution.Decompositioncanbedoneaccordingto(i)time;(ii)availableresources(facilitiesormachines);(iii)productfamilies;(iv)geographicalareas.Someofthedecompositionsmaybedesignedinsuchawaythattheyareactivatedautomaticallybythesystemitselfandotherdecompositionsmaybedesignedinsuchawaythattheyhavetobeactivatedbytheuserofthesystem.Decompositionisusedinmediumtermmodulesaswellasinde-tailedschedulingmodules.Inmediumtermplanningthedecompositionisoftenbasedontimeand/oronproductfamily(thesemaybeinternaldecom-positionsactivatedbythesystemitself).Theusermayspecifyinamediumtermplanningprocessageographicaldecomposition.Inthedetailedschedul- 1868PlanningandSchedulinginSupplyChainsingprocessthedecompositionisoftenmachinebased(suchadecompositionmaybedoneinternallybythesystemorimposedbytheuser).Onetypeofdiscretizationfeaturemaybeusedwhenthecontinuousversionofaproblem(forexample,alinearprogrammingrelaxationofamorerealisticintegerprogrammingformulation)doesnotyieldresultsthataresufficientlyaccurate.Inordertoobtainmoreaccurateresultscertainconstraintsmayhavetobeimposedongivenvariables.Forexample,productionquantitiesareoftennotallowedtoassumejustanyvalues,butonlyvaluesthataremultiplesofgivenfixedamountsorlotsizes(e.g.,thecapacityofatankinthebrewingofbeer).Thequantitiesthathavetobetransportedbetweentwofacilitiesalsohavetobemultiplesofafixedamount(e.g.,thesizeofacontainer).Thistypeofdiscretizationfeaturemaytransformacontinuousoptimizationproblem(i.e.,alinearprogram)intoadiscreteoptimizationproblem.Anothertypeofdiscretizationcanbedonewithrespecttotime.Itallowstheuserofthesystemtodeterminethesizeofthetimeunit.Iftheuserisonlyinterestedinaroughplan,hemaysetthetimeunittobeequaltoaweek.Thatis,theresultsoftheoptimizationthenonlyspecifywhatisgoingtobeproducedthatweek,butwillnotspecifywhatisgoingtobeproducedwithineachdayofthatweek.Iftheusersetsthetimeunitequaltooneday,theresultwillbesignificantlymoreprecise.Besidesspecifyingthesizesofthetimeunitsasystemmayusedifferenttimeunitsindifferentperiods.Thediscretizationfeatureisoftenimplementedinthemediumtermplanningmodules.Thefirstweekofathreemonthplanningperiodmaybespecifiedonadailybasis,thenextthreeweeksmaybedeterminedonaweeklybasis,andallactivitiesbeyondthefirstmonthareplannedbasedonacontinousmodel.Thistypeofdiscretizationdoesnotchangethenatureoftheproblem;iftheoriginalproblemisalinearprogram,thenitremainsalinearprogram.TheAPOsystemofSAPGermanyenablesthemodelertoactivateanddeactivatethediscretizationofvarioustypesofconstraintsinordertoimprovetheperformanceoftheoptimizationprocess.Forexample,discretizationmaybeusedfordailyandweeklytimebuckets,butnotformonthlytimebucketsinwhichLinearProgrammingisusedwithoutdiscretization.8.4AMediumTermPlanningModelforaSupplyChainThissectionconsidersatypicalmediumtermplanningmodelforasupplychain.Itdoesnotpresentthemodelinitsfullgenerality;thenotationneededthenotationadescriptionofthemodelisgivenwithmanyoftherelevantparametershavingfixedvalues.Italsodoesnotincorporateallofthefeaturesdescribedintheprevioussection(e.g.,allthetimeunitsareofthesamesize).Considerthreestagesinseries.Thefirstandmostupstreamstage(Stage1)hastwofacilities(orfactories)inparallel.TheybothfeedintoStage2whichisadistributioncenter(DC).BothStages1and2candelivertoaforamoregeneralmodelissimplytoocumbersome.Inordertosimplify 8.4AMediumTermPlanningModelforaSupplyChain187D.C.Factory 1Factory 2CustomerFig.8.5.AsystemwiththreestagescustomerwhichispartofStage3,seeFigure8.5.Thefactoriesdonothaveanyroomtostorefinishedgoodsandthecustomerdoesnotwanttoreceiveanyearlydeliveries.Theproblemhasthefollowingparametersandinputdata.Thetwofac-toriesworkaroundtheclock;sotheweeklyproductioncapacityavailableis24×7=168hours.TherearetwomajorproductfamiliesF1andF2.Asstatedbefore,inthemediumtermplanningprocessalltheproductswithinafamilyareconsideredidentical.Thedemandforecastsforthenext4weeksareknown(thetimeunitbeing1week).Inthissection,thesubscriptsandsuperscriptshavethefollowingmeaning.Thesubscriptt(t=1,...,4)referstoweekt.Thesubscripti(i=1,2),referstofactoryi.Thesubscriptj(j=1,2),referstoproductfamilyj.Thesubscriptl(l=1,2,3)referstostagel;l=1referstothetwofactories,l=2referstothedistributioncenter,andl=3referstothecustomer.Thesuperscriptpreferstoaproductionparameter.Thesuperscriptmreferstoatransportation(i.e.,moving)parameter.Thedemandforproductfamilyj,j=1,2,attheDClevel(Stage2)bytheendofweekt,t=1,...,4,isdenotedbyD2jt.Thedemandforproductfamilyj,j=1,2,atthecustomerlevel(Stage3)bytheendofweekt,t=1,...,4,isdenotedbyD3jt.Productiontimesandcostsaregiven:cpij=thecosttoproduceoneunitoffamilyjinfactoryi.ˆpij=thetime(inhours)toproduceoneunitoffamilyjinfactoryi.Theˆpijisjustanestimateoftheaveragetimerequiredtoproduceoneunit,sinceitcombinesprocessingtimeswithsetuptimes.Becausetherunlengthshavenotbeendeterminedyet,itisnotclearyetwhattheaverageproduction 1888PlanningandSchedulinginSupplyChainstimewillbe.Theˆpijisthereciprocaloftherateofproduction,seeChapter7.Holdingcostsandtransportationdatainclude:h=theweeklyholding(storage)costintheDCforoneunitofanytype.cmi2◦=thecostofmovingoneunitofanytypefromfactoryitotheDC.cmi◦3=thecostofmovingoneunitofanytypefromfactoryitothecustomer.cm◦23=thecostofmovingoneunitofanytypefromtheDCtothecustomer.τ=thetransportationtimefromanyoneofthetwofactoriestotheDC,fromanyoneofthetwofactoriestothecustomer,andfromtheDCtothecustomer;alltransportationtimesareassumedtobeidenticalandequalto1week.Thefollowingweightsandpenaltycostsaregiven:wj=thetardinesscostperunitperweekforanorderoffamilyjproductsthatarrivelateattheDC.wj=thetardinesscostperunitperweekforanorderoffamilyjproductsthatarrivelateatthecustomer.ψ=thepenaltyforneverdeliveringoneunitofproduct.Theobjectiveistominimizethetotaloftheproductioncosts,holdingorstoragecosts,transportationcosts,tardinesscosts,andpenaltycostsfornon-deliveryoverahorizonoffourweeks.InordertoformulatethisproblemasaMixedIntegerProgramthefollowingdecisionvariableshavetobedefined:xijt=numberofunitsoffamilyjproducedatfactoryiduringweekt.yi2jt=numberofunitsoffamilyjtransportedfromfactoryitotheDCinweektyi3jt=numberofunitsoffamilyjtransportedfromfactoryitocustomerinweektzjt=numberofunitsoffamilyjtransportedfromtheDCtocustomerinweekt.q2j0=numberofunitsoffamilyjinstorage(beingheld)attheDCattime0.q2jt=numberofunitsoffamilyjinstorage(beingheld)attheDCattheendofweekt.v2jt=numberofunitsoffamilyjthataretardy(havenotyetarrived)attheDCinweekt.v2j4=numberofunitsoffamilyjthathavenotbeendeliveredtotheDCbytheendoftheplanninghorizon(theendofweek4).v3j0=numberofunitsoffamilyjthataretardy(havenotyetarrived)atthecustomerattime0.v3jt=numberofunitsoffamilyjthataretardy(havenotyetarrived)atthecustomerbytheendofweekt.v3j4=thenumberofunitsoffamilyjthathavenotbeendeliveredtothecustomerbytheendoftheplanninghorizon(theendofweek4). 8.4AMediumTermPlanningModelforaSupplyChain189Notethatifyi2jt(yi3jt)aretransportedinweektfromfactoryitotheDC(customer),thentheyleavethefactoryinweektandarriveattheirdestinationinweekt+1.Thesameistruewithregardtothevariablezjt.TherearevariousconstraintsintheformofupperboundsUBiljandlowerboundsLBiljonthequantitiesoffamilyjtobeshippedfromfactoryitostagel.Theintegerprogramcannowbeformulatedasfollows:minimize4t=12j=12i=1cpijxijt+4t=12j=12i=1cmi2◦yi2jt+4t=12j=12i=1cmi◦3yi3jt+4t=12j=1cm◦23zjt+4t=12j=1hq2jt+3t=12j=1wjv2jt+3t=12j=1wjv3jt+2j=1ψv2j4+2j=1ψv3j4subjecttothefollowingweeklyproductioncapacityconstraints:2j=1ˆp1jx1jt≤168t=1,...,4;2j=1ˆp2jx2jt≤168t=1,...,4;subjecttothefollowingtransportationconstraints:y1ljt≤UB1ljt=1,...,4;y1ljt≥LB1ljory1ljt=0t=1,...,4;y2ljt≤UB2ljt=1,...,4;y2ljt≥LB2ljory2ljt=0t=1,...,4;3l=2yiljt=xijtt=1,...,4;j=1,2;i=1,2;2i=1yi3jt+zjt≤D3,j,t+1+v3jtt=1,...,3;j=1,2;zj1≤max(0,q2j0)j=1,2;zjt≤q2,j,t−1+y1,2,j,t−1+y2,2,j,t−1t=2,3,4;j=1,2;subjecttothefollowingstorageconstraints: 1908PlanningandSchedulinginSupplyChainsq2j1=max(0,q2j0−D2j1−zj1)j=1,2;q2jt=max(0,q2,j,t−1+y1,2,j,t−1+y2,2,j,t−1−v2,j,t−1−D2jt−zjt)j=1,2;t=2,3,4;subjecttothefollowingconstraintsregardingnumberofjobstardyandnum-berofjobsnotdelivered:v2j1=max(0,D2j1−q2j0)j=1,2;v2jt=max(0,D2jt+v2,j,t−1+zjt−q2,j,t−1−y1,2,j,t−1−y2,2,j,t−1)j=1,2;t=2,3,4;v3j1=max(0,D3j1)j=1,2;v3jt=max(0,D3jt+v3,j,t−1−zj,t−1−y1,3,j,t−1−y2,3,j,t−1)j=1,2;t=2,3,4.Itisclearthatmostvariablesinthismixedintegerprogrammingformu-lationarecontinuousvariables.However,thetransportationvariablesyiljtaresubjecttodisjunctiveconstraints.Adifferentformulationoftheproblemcouldhavesomeinteger(0−1)variablestomakesurethatthecontinuous(transportation)variablesyiljtareeither0orlargerthanthelowerboundLBilj.Moreover,notethatthoseconstraintsinwhichavariableisequaltothemaxofanexpressionand0arenonlinear.Inordertoensurethatthegivenvariableremainsnonnegative,anadditionalbinaryvariablehastobeintroduced.Notethatarelaxationoftheformulationdescribedabovewithoutthedisjunctiveconstraintsprovidesavalidlowerboundonthetotalcost.Thefollowingnumericalexampleillustratesanapplicationofthemodeldescribedabove.Example8.4.1(MediumTermPlanning).Considerthefollowingin-stanceoftheproblemdescribedabove.Theproductiontimesandcostsinfactory1are:ˆp11=0.001hours(i.e.,3.6sec.)andˆp12=0.002hours(i.e.,7.2sec.);cp11=$1.00andcp12=$0.50.Theproductiontimesandcostsconcerningfactory2are:ˆp21=0.002hours(i.e.,7.2sec.)andˆp22=0.003hours(i.e.,10.8sec.);cp21=$0.50andcp22=$0.25.TheholdingcostforoneunitofanytypeofproductattheDC(h)is$0.10perunitperweek.Thetransportationcostsarecm12◦=$0.10perunit;cm22◦=$0.30perunit;cmi◦3=$0.05fori=1,2,cm◦23=$0.50perunit.TheforecastdemandattheDCandfromthecustomerforthetwodifferentproductfamiliesarepresentedinthetablebelow.week1week2week3week4D21t20,00030,00015,00040,000D22t050,00030,00050,000D31t10,0005,00015,00040,000D32t010,00005,000 8.4AMediumTermPlanningModelforaSupplyChain191Theweeklyshipmentsofproductfamily1fromfactory1totheDChavetocontainatleast10,000unitsorotherwisethereisnoshipment,i.e.,LB121=10,000.Fromfactory2totheDCtherehastobeeachweekatmostashipmentof10,000unitsofproductfamily2,i.e.,UB222=10,000.Thetransportationtimeτis1week.Thetardinesscostw1(w2)is$70.00($35.00)perunitperweek.Thetardinesscostw1(w2)is$140.00($105.00)perunitperweek.Thepenaltycostψfornotdeliveringatallis$1000.00perunit.RunningthesedatathroughaMixedIntegerProgrammingsolver(assum-ingthattheboundaryconditionsv3j0andq2j0arezero)yieldsthefollowingproductionandtransportationdecisions.week1week2week3week4x11t0000x12t47,33320,00050,0000x21t65,00033,50076,5000x22t12,66710,0005,0000week1week2week3y121t000y131t000y221t50,00018,50036,500y231t15,00015,00040,000y122t47,33320,00050,000y132t000y222t2,66710,0000y232t10,00005,000Thetotalcostofthissolutionisapproximately$3,004,750.00.(Theexacttotalcostmaydependonwhetherornotintegralityassumptionsconcerningthequantitiesproducedandtransportedareineffectaswellasonothersettingsintheprogram.)Ifanadditionalconstraintisaddedtothisproblemrequiringtheproduc-tionlotsizestobemultiplesof10,000,thenweobtainthefollowingsolution.week1week2week3week4x11t0000x12t60,00020,00060,0000x21t70,00030,00080,0000x22t010,00000 1928PlanningandSchedulinginSupplyChainsweek1week2week3y121t000y131t000y221t55,00015,00040,000y231t15,00015,00040,000y122t50,00020,00055,000y132t10,00005,000y222t010,0000y232t000Thetotalcostinthiscaseisindeedhigherthanthetotalcostwithouttheproductionconstraintthatrequiresitemstobeproducedinlotsof10,000.Thetotalcostisapproximately$3,017,000.00.Theincreaseislessthan0.5%.Theincreasedcostsaremainlyduetoexcessproduction(thetotalproductionquantitiesnowexceedthetotaldemandquantities)and,asaconsequence,additionaltransportationandholdingcosts.Itisclearthattheformulationofthismediumtermplanningproblemcanbeextendedveryeasilytomoretimeperiods,morefactoriesatthefirststageandmoreproductfamilies.Anextensiontomorestagesmaybealittlebitmoreinvolvedwhenthereisanincreaseinthecomplexityofroutingpatterns.8.5AShortTermSchedulingModelforaSupplyChainTheshorttermschedulingproblemforafacilityinasupplychaincanbedescribedasfollows:Theoutputofthemediumtermplanningproblemspec-ifiesthatovertheshorttermnjitemsoffamilyjhavetobeproduced.Theschedulingproblemcaneitherbemodeledasajobshop(orflexibleflowshop)thattakesalltheproductionstepsinthefacilityintoaccount,orasasome-whatsimplersingle(orparallel)machineschedulingproblemthatfocusesonlyonthebottleneckoperation.Iftheoperationsinafacilityarewellbalancedandthelocationofthebottleneckdependsonthetypesofordersthatareinthesystem,thentheentirefacilitymayhavetobemodeledasajobshop.Ifthebottleneckinthefacilityisapermanentbottleneck(thatnevermoves),thenafocusonthebottleneckmaybejustified.Ifthebottleneckstageismodeledasaparallelmachineschedulingmodel,thentheparallelmachinesmaynotbeidentical.Theymayalsobesubjecttodifferentmaintenanceandrepairschedules.Thereis,ofcourse,acloserelationshipbetweenthetimeˆpijinthemediumtermplanningprocessandtheprocessingtimeofanorderintheshorttermdetailedschedulingproblem.Theˆpijinthemediumtermplanningprocessisanestimateandmaybeavalueanywhereinbetweentheaverageprocessing 8.5AShortTermSchedulingModelforaSupplyChain193timeofanorderatthebottleneckoperationandthetotal(estimated)throug-puttimeofanorderthroughthefacility.Theˆpijisafunctionoftheexactprocessingtimes,thesequencedependentsetuptimes,andtherunlenghts.Intheremainingpartofthissectionthesubscriptiwillrefertoa”ma-chine”insteadofafactoryandthesubscriptjwillrefertoanorderorajobratherthanaproductfamily.Anordercannotbereleasedbeforeallthere-quiredrawmaterialhasarrived(thesedatesaretypicallystoredinaMaterialRequirementsPlanning(MRP)system).Thatis,orderjhasanearliestpossi-blestartingtimethatistypicallyreferredtoasareleasedaterj,acommittedshippingdatedjandapriorityfactororweightwj.Dependentupontheman-ufacturingenvironment,preemptionsmayormaynotbeallowed.Everytimeamachineswitchesoverfromonetypeofitemtoanothertypeofitemasetupcostmaybeincurredandasetuptimemaybeneeded.Ifaschedulecallsforalargenumberofpreemptions,alargenumberofsetupsmaybeincurred.Theobjectivetobeminimizedmayincludethetotalsetuptimesonthemachinesatthebottleneckaswellasthetotalweightedtardiness,whichisdenotedbywjTj(seeChapter5).Theobjectivemaybeformulatedasα1wjTj+α2Iijksijk,wheretheα1andtheα2denotetheweightsofthetwopartsoftheobjectivefunction.Thefirstpartoftheobjectivefunctionisthetotalweightedtardinessandthesecondpartoftheobjectiverepresentsthetotalofallsetups;theindicatorvariableIijkis1ifjobkfollowsjobjonmachinei,theindicatorvariableis0otherwise.Thisschedulingproblemmaybetackledviaanumberofdifferenttech-niques,includingacombinationofdispatchingrules,suchastheShortestSetupTime(SST)firstrule,theEarliestDueDatefirst(EDD)rule,andtheWeightedShortestProcessingTimefirst(WSPT)rule,seeChapter5andAppendixC.Othertechniquesmayincludegeneticalgorithmsorintegerprogrammingapproaches.Inthisphase,however,integerprogrammingap-proachesarenotoftenusedbecausetheyarecomputationallyquiteintensive.Example8.5.1(ShortTermScheduling).Considerthetwofactoriesde-scribedinthemediumtermplanningprocessintheprevioussection.Inthedetailedschedulingprocessthetwofactoriesmaybescheduledindependentlyfromoneanotherandtheschedulingisdoneoneweekatthetime.Considerfactory1withthetwoproductfamilies.Theproductionprocessinthisfac-toryconsistsofvarioussteps,butoneofthesestepsisthebottleneck.Thisbottleneckconsistsofanumberofresourcesinparallel.Considerinthesamewaytheoperationsoffactory2initsfirstweekofoperations.Thesolutionoftheintegerprogramyieldsx21t=65,000andx22t=12,667.Ofthe65,000ofproductfamily1atotalof50,000hastobeshippedtotheDCandtheremainderhastogotothecustomer.Ofthe12,667ofproductfamily2atotalof2667hastobeshippedtotheDCandtheremaining10,000hastogotothecustomer. 1948PlanningandSchedulinginSupplyChainsAssumethatforthisprocessthefollowingmoredetailedinformationisavailable(whichwasnottakenintoaccountintothemediumtermplanningprocess).Thetimeunitintheschedulingprocessis1hour;thisisincontrasttothe1weekinthemediumtermplanningprocess(inactualimplementationsthetimeunitintheschedulingprocesscanbemadearbitrarilysmall).Theschedulinghorizonis1week.Recallthat2hoursofthebottleneckresourcearerequiredtoproduce1000unitsoffamily1infactory2,whereas3hoursofthebottleneckresourcearerequiredfor1000unitsoffamily2.Thisimpliesthat,basedontheseestimatedproductiontimes,theplannedproductiontakesthefullcapacityofthebottleneckresource(inhours)65×2+12.667×3=168.However,the2and3hoursrequirementofthebottleneckresourceareonlyestimates.Theyareestimatesthatarebeingusedinthemediumtermplanningprocessinordernottohavetomakeadistinctionbetweensequencedependentsetuptimesandruntimes.Theactualruntimes(orprocessingtimes),excludinganysetuptimesarethefollowing:Toproduceinfactory21000unitsoffamily1,1.75hoursofthebottleneckresourceisrequired,whereas1000unitsoffamily2requires2.5hoursofthebottleneckresource.Tostartproducingunitsoffamily1asetupof16hoursisrequired.Tostartproducingunitsoffamily2asetupof6hoursisrequired.Ifeachoneoftheproductsweretobeproducedinasingleruninthatweek,thentheentireproductioncouldbedonewithin168hours,since16+65×1.75+6+12.66×2.5=167.4So,iftherearenottoomanysetuptimestheoriginalassumptionsforthemediumtermplanningmodelareappropriate.However,theshipmenttothecustomerissupposedtogoonatruckattime120(after5days),whereastheshipmenttotheDCtakesplaceattheendoftheweekattime168.Alltherawmaterialrequiredtoproducefamily1productsareavailableattime0,whereasthematerialnecessarytoproducefamily2productsareonlyavailableafter2days,i.e.,attime48.Thisproblemcanbemodeledasasinglemachineschedulingproblemwithjobshavingdifferentreleasedatesandbeingsubjecttosequencedependentsetups.Theobjectiveisα1Cmax+α2wjTj.Thereare4differentjobs,withthefollowingprocessingtimes,releasedates,andsequencedependentsetuptimes.Eachjobischaracterizedbyitsfamilytypeanditsdestination.Jobs1and2arefromthesamefamily,sothereisazerosetuptimeifonejobisfollowedbytheother.Ifeitherjob1orjob2followsjob3or4,thenasetupof16hoursisrequired.Ifjob3or4followsjob1or2asetupof6hoursisrequired. 8.6CarlsbergDenmark:AnExampleofaSystemImplementation195job1234pj87.526.256.6725rj003636dj168120168120Twoschedulingapproachesmaybeappealing.OnewouldschedulethejobsaccordingtotheShortestSetupTimefirstrule(withtiesbrokenaccordingtotheEarliestDueDaterule).Thisapproachwouldyieldtheschedule1,2,4,3;afterjob1hasbeencompletedjob2hastostartbecauseithasazerosetuptime.Alljobsarecompletedbytime168.However,job4iscompletedlate.Ithadtobeshippedbytime120anditisshippedbytime161.ThesecondapproachfollowstheEarliestDueDatefirstrule(withtiesbrokenaccordingtotheShortestSetupTimefirstrule).Thisapproachyieldsschedule2,4,3,1.Sincethereisanadditionalsetuptimethemakespanis167.4+16=183.4.Theshipmenttothecustomerleavesontime,buttheshipmenttotheDCleaveslate.Theweightsα1andα2intheobjectivefunctiondeterminewhichscheduleismorepreferable.Theresultsfromthedetailedschedulinganalysismay,forvariousreasons,notbeuseful.Whentryingtominimizethemakespan(inordertoensurethattheproductionoftherequiredquantitiesarecompletedwithinthatweek),itmayturnoutthattheredoesnotexistaschedulethatcancompletetherequestedproductionwithinaweek.Thereasonmaybethefollowing:theproductiontimesˆpijthatwereenteredinthemediumtermplanningprob-lemweremereestimatesbasedonfactorydata,includingaverageprocessingtimesonbottleneckmachines,expectedthroughputtimes,expectedsetuptimes,andsoon.However,thevalueˆpijdidnotrepresentanaccuratecycletime,sincetheaverageproductiontimemaydependontherunlengthofthebatchesatthebottleneck.Itmaybethattheschedulegeneratedinthede-tailedschedulingprocesshasbatchsizesthatareveryshortwithanaverageproductiontimethatislargerthantheestimatesusedinthemediumtermplanningprocess.Ifthereisamajordiscrepancy(i.e.,thefrequencyofthesetupsisconsiderablyhigherthanusual),thenanewestimatemayhavetobedevelopedfortheˆpijinthemediumtermplanningprocessandtheintegerprogrammingproblemhastobesolvedagain.8.6CarlsbergDenmark:AnExampleofaSystemImplementationTherearemanysoftwarevendorsthatsellcustommadesolutionsforsupplychainplanningandscheduling.OneofthelargestcompaniesinthisfieldisSAP,whichisbasedinWalldorf(Germany).SAPhasadivisionthatdevelopsitsso-calledAdvancedPlannerandOptimizer(APO)system;thissupply 1968PlanningandSchedulinginSupplyChainsOptimizerLP solverConstraint-based programmingGenetic algorithmsGO!Live CacheModel GeneratorModel CheckerMeta-HeuristicsControls general strategyFig.8.6.TheSAP-APOoptimizerarchitecturechainplanningandschedulingsystemhasfunctionalitiesatvariouslevels,includingthetacticallevelandtheoperationallevel.Onthetacticallevel,mediumtermplanningscenarioscanbemonitoredforaglobalchainfromdistributioncenterstoplantsandsuppliers.Theoptimizerautomaticallyprocessesbillsofmaterialswhiletakingcapacitiesintoaccount,anditminimizestransportationcosts,productioncosts,andholdingorstoragecosts.Thesheercomplexityofthisglobalviewishandledthrougharough-cutmodelthataggregatestimeunitsinbuckets(e.g.,dayorweek)andproductsandresourcesinfamilies.Ontheoperationallevel,APOreliesonadetailedschedulingmodel.Atthisleveltheshortterm,daytodayoperationsaremonitored,takingintoaccounttheidiosyncrasiesofalltheoperationsinthesupplychain.Theop-timizerschedulestheorderstakingintoaccountalltherulesandconstraintsthatareprevalentinacomplexmanufacturingenvironment(whichmaybeamulti-stageproductionprocesswithprimaryresources,secondaryresources,andalternativeroutings).Figure8.6showsthearchitectureoftheoptimizerinAPO.ForlongtermandmediumtermplanningAPOusesitsLPsolvers(CPLEX).APOhasvariousapproachesforshorttermplanninganddetailedscheduling,includingConstraintProgramming,GeneticAlgorithms,andRepairAlgorithms. 8.6CarlsbergDenmark:AnExampleofaSystemImplementation197ThissectiondescribesanimplementationoftheSAP-APOsystematthebeerbrewerCarlsbergA/SinDenmark.Themodelingthatformsthebasisforthiscaseissomewhatsimilartothemodelsdescribedinthefourthandfifthsectionofthischapter.CarlsbergDenmarkA/S,thelargestbeerbrewerinScandinavia,startedin2001asupplychainprojectwiththeobjectivetodecreaseinventorycosts,tooptimizesourcingdecisions,toincreasecustomerservicelevelandingeneraltochangethebusinesstoamoredemanddrivenprocess.CarlsbergselectedAPO.Theprojectisanexampleofhowareal-lifeimplementationtakesplanningandschedulingissuesinvariousstagesofasupplychainintoaccount.Thesystemhasbeenoperationalsincetheendof2002.Thesupplychainconsideredintheprojectconsistsofthreestages.Thefirststageistheproductionprocessofthebeerattwobrewerieswith2and4fillinglines,respectively.Eachfillinglinehasadifferentcapacity.Thesecondstageconsistsofadistributioncenter(DC)andthethirdstageconsistsofthelocalwarehouses,seeFigure8.5.Inthefirststagetherearethreeproductionsteps,namelybrewing(andfermentation),filteringandfillingofthebeer.Allthreestepshavealimitedcapacity,butthebottleneckisusuallythefillingstep.Theresourcesforthefillingoperationsatthetwoplantshavedifferentcostsandprocessingtimes.Whencreatingtheproductionordersforbrewingandfilling,differentlotsizeconstraintshavetobetakenintoaccount.Productionordersforthebrewinghavealwaysafixedlotsizebecausethebrewingtankhastobefilled.Ifthedemandquantityishigherthanthefixedlotsize,thenadditionalproductionordershavetobecreatedforthebrewingprocess(eachwiththefixedlotsizeastheproductionquantity).Ordersbelowtheminimallotsizeareincreasedtotheminimallotsizeandordersabovetheminimallotsizeareeitherroundedupordowntotheclosestintegervalue.Thefillingresourceshavetobefilledupto100%.Thereisfurtherasplitinthebusinessprocessesaccordingtothesalesvolumesofthevariousproducts.Therearethreecategories:A,B,andC.CategoryAarethefastmoversandincludethewell-knownbrandsCarlsbergPilsandTuborgGreen.CategoryCarethe(moreexpensive)slowmovers.Oncethebeerisbottled,ithastobetransportedeithertothedistributioncenter(DC)ortoalocalwarehouse.Dependingonthedifferentproductsandthequantitiestobetransported,eitheradirectdeliveryfromtheplanttoalocalwarehouseoratransportviatheDCisbetter.Again,lotsizeconstraintshavetobetakenintoconsiderationwhencreatingtransportorders.Thetransportdurationsdepend,ofcourse,ontheoriginandthedestination.OneofthemainobjectivesofCarlsbergistoprovideagivenlevelofservicetoitscustomers.Atypicalwaytoachieveagivenservicelevelistokeepsafetystocksatthewarehouses.Thehigherthesafetystocklevels,thehighertheservicelevels,butalsothehighertheinventorycosts.Onefunctionofasupplychainmanagementsystemisthecomputationofthelowestlevelsofsafetystocksthatachievethedesiredservicelevels.Carlsbergusesadvancedsafetystockmethodstocomputesafetystockvaluesforallitsproductsatits 1988PlanningandSchedulinginSupplyChainsDCaswellasatitslocalwarehouses.Thesesafetystocklevelsdependonthegivenservicelevel,thedemandforecast,theuncertaintyintheforecast,thereplenishmentleadtimeandthetypicallotsizes.Themediumtermplanningmoduleplansaheadfor12weeks,withthefirst4weeksindaysandtheremaining8weeksinweeklyperiods.Assumingagivendemandpattern(salesordersandforecasts),APOcreatesaMixedIntegerProgram,alongthelinesdescribedintheprevioussection,andtriestofindasolutionwithminimumcost.Thetotalcostsincludeproductioncosts,storagecosts,transportationcosts,latedelivery(tardiness)costs,non-deliverycosts,andviolationofthesafetystocklevelscomputedinthefirststep.Someofthecostsmentionedabovecanbespecifiedinanexactway,suchastheproductionandtransportationcosts.Othercosts,suchasstorage,violationofsafetystock,andlateandnon-deliverycosts,merelyrepresenttheprioritiesofCarlsberg.If,forexample,CarlsbergconsidersthesafetystockinthelocalwarehousesmorecriticalthanthesafetystockintheDC,thenthecostassignedtotheviolationofsafetystockforaproductattheDCislessthanthecostsofviolatingsafetystocksofthesameproductsatthelocalwarehouses.Ifneithersafetystockcanbemaintained,thenthesystemwillcreateatransportfromtheDCtothelocalwarehouse(providedthedifferencebetweenthecostsofsafetystockviolationsattheDCandatthelocalwarehouseishigherthanthetransportationcostfromtheDCtothelocalwarehouse).Clearly,allcosttypesarestronglyrelatedwithoneanotherandmodifyingonetypeofcostcanhavemanyunforeseenconsequencesinthesolutiongenerated.Carlsbergdevelopeditsowncostmodelforstoragecosts;thismodel,forexample,takesintoaccountthelocationoccupiedbyapallet,themaximumnumberoflevelspalletscanbestacked,thenumberofproductsperpallet,andthewarehouseitself.Basedontheseparametersforeachproductateachlocation,storagecostscanbecomputed.Thefollowingconstraintshavetobetakenintoconsideration,namelytheproductiontimesinthethreeproductionsteps,thecapacityofthebottlingre-sourcesonadailyorweeklylevel,thetransportationtimesbetweenlocations,thelotsizeconstraints,theexistingstockandtheresourceconsumptions.Themediumtermplanistheresultofvariouscoststrade-offsandma-terialconsumption.Thesystemgeneratesforthenext12weekstheplannedproductionquantitiesforthethreeproductionstepsindetail(includingthequantityofeachproducttobebottledoneachfillingresourceaswellasthequantitiestobetransportedfromonelocationtoanother.Theshorttermschedulingstartsitscomputationsusingresultsobtainedfromthemediumtermplan.Theplannedproductionordersforthefirstweekthatcomeoutofthemediumtermplanningsystemaretransformedintoshorttermproductionordersonwhichadetailedschedulingprocedurehastobeapplied.Theseproductionordersarethenscheduledonthefillingresourcesbyapplyingageneticalgorithmwithasobjectivetheminimizationofthesumofthesequencedependentsetuptimesandthesumofthetardinesses.Theduedatesarespecifiedbythemediumtermplanningproblemandareequaltothe 8.7Discussion199startingtimesofthetransportationorders.Itispossiblethattheresultsofthemediumtermplanarechangedbytheshorttermschedulingprocedure(i.e.,adifferentfillingresourcemaybeselectedinthesameplant).Afterthedetailedschedulinghasbeencompleted,thetransportationplanningandschedulinghastobedone.Inthisstepthetrucksthatprovidethetransportationbetweenthedifferentlocationsareloadedbasedontheresultsthatcomeoutofthemediumtermplanandtheresultsfromthedetailedschedulingprocedures.Inordertomaximizetheutilization,atruckmaytransportonanygiventripvariousdifferentproducts.Anewmediumtermplanisgeneratedeveryday.Thedailyruntakesintoaccountthemostup-to-datecapacitysituationofalltheavailableresources,theresultsofthepreviousdaydetailedschedule,andthemostcurrentdemandforecast.Afterwards,anewdetailedscheduleandtransportationplanaregenerated.ThegenerationofthemediumtermplanissplitintothreeMixedInte-gerPrograms,whicharesolvedinconsecutiveruns.EachMIPhasbetween100,000and500,000variablesandbetween50,000and150,000constraints.Totalrunningtimeisabout10-12hours.EachMIPusesproductdecomposi-tionmethods,whichcreates5to10subproblemseach.Thegenerationofthesubproblemshastotakeintoaccountdifferentprioritiesforthefinishedbeerproductsandthefactthatthesamebrewedandfilteredbeertypemayendupindifferentendproducts.Thequalityofthesolutionismeasuredinbusinesstermsaswellasintechnicalterms.Onlybyconsideringalldimensionsonecanspeakofa”good”ora”bad”solution.Thetechnicaldataandmeasurestendtobeeasytocollectandunderstand;themoreimportantbusinessmea-suresarehardertounderstandandverify.Thetwomostimportanttechnicalmeasuresare(i)thedifferencebetweenthecostsoftheMIPsolutionandtheLPrelaxationsolution,and(ii)thedifferencebetweentheoveralldeliverypercentagesoftheMIPsolutionandtheLPrelaxationsolution.Thediffer-encebetweenthetwocostsisonaveragebetween0.2and10%,butsometimesitreaches400%.AhugecostdifferencebetweentheMIPandtheLPrelax-ationcanoccurwhentheLPcanfulfillalldemandswhiletheMIPcannot(becauseofthelotsizeconstraints).Asunfilleddemandbringsaboutaveryhighpenaltycost,thecostdifferencebetweentheMIPandtheLPrelaxationmaythenbeveryhigh.Theuserinterfacesofthesystemsare,ofcourse,quiteelaborateandin-cludethetypicalGanttcharts,seeFigures8.7and8.8.8.7DiscussionThepurposeofthischapteristoprovideinsightsintotheuseofplanningandschedulingmodelsinsupplychainmanagement,aswellasintotheinformationsharingandinteractionsthatoccurbetweenthedifferenttypesofmodelsthatformthebasisforasystem.Intheliterature,planningmodelshaveoftenbeen 2008PlanningandSchedulinginSupplyChainsFig.8.7.Carlsberg-Denmarkplanningsystemuserinterfaceanalyzedindetail;schedulingmodels,ontheotherhand,havebeenstudiedlessoftenwithinasupplychainmanagementframework.Theinteractionsandinformationsharingbetweentheplanningmodelsandtheschedulingmodelsalsodeservemoreattention.Thereareseveralreasonswhyitisnotthateasytoincorporateplanningsystemsandschedulingsystemsinoneframework.Onereasonisthatintheplanningstagetheobjectiveismeasuredindollarterms(e.g.,minimizationoftotalcost),whereasintheschedulestagetheobjectiveistypicallymeasuredintimeunits(e.g.,minimizationoftotaltardiness).Asecondreasonisthatthetimeperiodsoverwhichtheplanningmoduleandtheschedulingmoduleoptimizemayoverlaponlypartially,seeFigure8.3.Thehorizonoverwhichtheschedulingmoduleoptimizestypicallystartsatthecurrenttimeandcoversarelativelyshortterm.Theplanningmoduleoptimizesoveraperiodthatstartsatsometimeinthefuture(sinceitmayassumethatallschedulesbeforethispointintimealreadyhavebeenfixed)andcoversalongterm.Theunitsoftimeusedinthetwomodulesmaybedifferentaswell.Intheschedulingmoduletheunitmaybeanhouroraday;intheplanningmoduleitmaybeaweekoramonth.Comparingthemodelingthatisbeingdoneinpracticeformediumtermplanningprocesseswiththemodelsthathavebeenstudiedintheresearchliterature,itbecomesclearthattherearedifferencesinemphasis.Whenmulti-stagemodelsareconsideredintheplanningandschedulingresearchliterature,thereismoreofanemphasisonsetupcosts(typicallysequenceindependent) Exercises201Fig.8.8.Carlsberg-Denmarkschedulingsystemuserinterfaceandlessofanemphasisontransportationcosts;inthemodelingthatisdoneinpractice,thereisaverystrongemphasisontransportationcostsandlessofanemphasisonsetupcosts.Incorporatingbothsetupcostsandtransportationcostsinamulti-stageplanningmodelmaycausethenumberofvariablestobecomeprohibitivelylarge.Exercises8.1.ConsiderthemodelinSection8.4.Howdothenumberofvariablesandconstraintsdependonthenumberofcustomers?8.2.ConsiderthemodelinSection8.4.(a)Howdothenumberofvariablesandthenumberofconstraintsincreasewhenthenumberoffactoriesinstage1isincreasedfrom2to3.(b)Howdothenumberofvariablesandconstraintsincreaseifthenumberofdistributioncentersisincreasedfrom1to2(assumethatbothdistributioncenterscanreceivefrombothfactoriesandcandelivertothesamecustomer). 2028PlanningandSchedulinginSupplyChains8.3.ConsiderExample8.4.1.(a)SolvethesameinstancebutassumenowthatLB112=0.(UseeitherIlog’sOPLcodeorDashOptimization’sMoselcodeforthisproblemontheaccompanyingCD-ROM.)ComparethenewsolutionwiththesolutioninExample8.4.1.(b)Ifthislowerboundiszero,doestheproblemreducetoalinearpro-grammingproblem?Explainyouranswer.8.4.ConsiderExample8.4.1.Solvethesameinstanceagain,butassumenowthatLB222=LB223=5000.ComparethesolutionobtainedwiththesolutionofExample8.4.1.8.5.ConsiderExample8.4.1.Solvethesameinstance(againuseeithertheDashcodeortheOPLcodeontheCD),butnowassumethattheproductionlotsizeis:(a)20,000,and(b)30,000.(c)Doesthetotalcostincreaseasafunctionoftheproductionlotsizeinalinearmanner,concavely,orconvexly?Giveanexplanationforyourconclusion.8.6.ModifyeithertheDashcodeortheOPLcodeontheCDforExample8.4.1insuchawaythatthetransportationquantitiesarealwaysmultiplesofafixedvalueK(i.e.,theydonothavejustalowerboundK).8.7.UsethecodedevelopedinExercise8.6tosolvetheinstanceswithtrans-portationquantitiesthataremultiplesof(a)10,000;(b)20,000;(c)30,000.(d)Doesthetotalcostincreaseasafunctionofthetransportationquantityinalinearmanner,concavely,orconvexly?Giveanexplanationforyourconclusion.8.8.Comparethesensitivityofthetotalcostwithrespecttotheproductionlotsizetothesensitivitywithrespecttothetransportationquantity.8.9.Formulateanintegerprogramforamodelwithtwodistributioncentersinthesecondstage.Eachdistributioncenterhasitsowncustomer(i.e.,eachoneofthetwocustomerscanonlybesuppliedbyoneofthedistributioncenters).ComparethenumberofvariablesandthenumberofconstraintsinthisMixedIntegerProgramwiththenumberofvariablesandconstraintsintheoriginalformulation.8.10.Formulateanintegerprogramforamodelwithtwofactories,threeproducts,twodistributioncentersinthesecondstageandonecustomersinthethirdstage.Eachdistributioncentercansupplythecustomer.Compare CommentsandReferences203thenumberofvariablesandthenumberofconstraintsinthisMixedIntegerProgramwiththenumberofvariablesandconstraintsintheoriginalformu-lation.CommentsandReferencesThereisanextensiveliteratureonsupplychainmanagement.Manypapersandbooksfocusonsupplychaincoordination;however,asignificantamountofthisworkhasanemphasisoninventorycontrol,pricingissues,andthevalueofinformation,seeHaxandMeal(1975),BowersoxandCloss(1996),Simchi-Levi,KaminskyandSimchi-Levi(2000),ChopraandMeindl(2001),andStadtlerandKilger(2002).AfairamountofresearchhasbeendoneonthesolutionmethodsapplicabletoplanningandschedulingmodelsforSupplyChainManagement,seeMuckstadtandRoundy(1993),Shapiro(2001),andMiller(2002).Someoftheplanningandschedulingmodelshavebeenstudiedinaratherrestrictedmannerinordertoobtaineleganttheoreticalresults;see,forexample,HallandPotts(2003).Someresearchhasbeendoneonmoreintegratedmodelsintheformofhierarchicalplanningsystems;thisresearchhasresultedinframeworksthatincorporateplanningaswellasschedulingmodels,seeBarbarosogluandOzgur(1999),Dhaenens-FlipoandFinke(2001).Onlyalimitedamountoftheresearchhasfocusedonthedetailsofactualap-plicationsandimplementationsofthemoreintegratedsystems,sincenotmanyoftheseinstallationshavebeensuccesful,seeHadaviandVoigt(1987),Hadavi(1998),ShepherdandLapide(1998),andSadeh,Hildum,andKjenstad(2003).Ex-amplesofplanningandschedulingapplicationsincontinuousmanufacturingcanbefoundinHaq(1991),Akkiraju,Keskinocak,MurthyandWu(1998),Murthy,Akkiraju,Goodwin,Keskinocak,Rachlin,Wu,Kumaran,Yeh,Fuhrer,Aggarwal,Sturzenbecker,JayaramanandDaigle(1999),Rachlin,Goodwin,Murthy,Akkiraju,Wu,Kumaran,andDas(2001),andKeskinocak,Wu,Goodwin,Murthy,Akkiraju,KumaranandDerebail(2002).ExamplesofplanningandschedulingindiscretemanufacturingaredescribedinArntzen,Brown,HarrisonandTrafton(1995),DeBontridder(2001)andVandaeleandLambrecht(2001).ThischapterismainlybasedonthepaperbyKreiplandPinedo(2004).Thethird,fourth,fifth,andsixthsectionofthischapterhavebeeninspiredprimarilybythearchitectureofsystemsdevelopedbySAPGermanyAG,seeBraun(2001),BraunandGroenewald(2000),andStrobel(2001).ThemodeldescribedinthefourthsectionisasimplifiedexampleofaclassofstandardplanningmodelsthathasbeenthebasisforanumberofactualimplementationsbySAPGermanyAGinvariousdifferentindustries,fromsemiconductor(DRAM)manufacturingcompaniestobeerbrewerswithelaboratedistributionsystems.TheOPLcodeforExample8.4.1ontheCD-ROMisduetoIrvinLustigfromILOG.TheMoselcodeforExample8.4.1ontheCD-ROMisduetoAlkisVaza-copoulosandNitinVermafromDashOptimization. PartIIIPlanningandSchedulinginServices9IntervalScheduling,Reservations,andTimetabling......20710SchedulingandTimetablinginSportsandEntertainment.23111Planning,Scheduling,andTimetablinginTransportation.25512PlanningandSchedulinginHealthCare................29113WorkforceScheduling................................317 Chapter9IntervalScheduling,Reservations,andTimetabling9.1Introduction.................................2079.2ReservationswithoutSlack....................2099.3ReservationswithSlack.......................2129.4TimetablingwithWorkforceConstraints.......2159.5TimetablingwithOperatororToolingConstraints..................................2189.6AssigningClassestoRoomsatU.C.Berkeley...2249.7Discussion...................................2269.1IntroductionSchedulingactivitiesinanenvironmentwithresourcesinparallelmayrequireattimesareservationsystem.Eachactivity(i.e.,reservation)issupposedtooccupyoneoftheresourcesforagiventimeperiod.Activityj,j=1,...,n,hasadurationpjandhastofitwithinatimewindowthatisspecifiedbyanearlieststartingtimerjandalatestterminationtimedj.Theremayormaynotbeanyslack,i.e.,eitherpjW),thenactivitiesjandkmaynotoverlapintime.Thistypeoftimetablingisinwhatfollowsreferredtoastimetablingwithworkforceorpersonnelconstraints.Inthesecondtypeoftimetablingproblemeachoperatorhasitsowniden-tityandisunique.(Anoperatormaynowbeequivalenttoaspecifictoolorfixturethatisrequiredinordertoperformcertainactivities.)Eachactivitynowrequiresaspecificsubsetoftheoperatorsand/ortools.Inorderforanactivitytobescheduledalltheoperatorsortoolsinitssubsethavetobeavailable.Twoactivitiesthatneedthesameoperatorcanthereforenotbeprocessedatthesametime.Thissecondtypeoftimetablingcanoccurinmanydifferentsettings.Con-sider,forexample,alargerepairshopforaircraftengines.Inordertodocertaintypesofrepairsitisnecessarytohavespecifictools,equipment,and
9.2ReservationswithoutSlack209operators.Agiventoolorpieceofequipmentmayberequiredforcertaintypesofrepairs;timetablingmaythereforebenecessary.Asecondexampleofthistypeoftimetablingoccurswhenmeetingshavetobescheduled.Theop-eratorsarenowthepeoplewhohavetoattendthemeetingsandeachmeetinghastobeassignedtoatimeperiodinwhichthosewhohavetoattendareabletodoso.Themeetingroomscorrespondtotheresources.Athirdexampleofthistypeoftimetablingoccurswhenexamshavetobescheduled.Eachoperatorrepresentsastudent(oragroupofstudents).Twoexamsthathavetobetakenbythesamestudent(orgroupofstudents)cannotbescheduledatthesametime.Theobjectiveistoschedulealltheexamswithinagiventimeperiod.Onereasonforcoveringreservationproblemsandtimetablingproblemsinthesamechapteristhatbothtypesofproblemsleadtowell-knowngraphcol-oringproblems.Thereservationproblemwithzeroslackandthetimetablingproblemwithoperatorortoolingconstraintsarebothcloselyrelatedtoawell-knownnodecoloringproblemingraphtheory.Thereservationmodelwithslackis,ofcourse,ageneralizationofthereservationmodelwithoutslack.Thetimetablingproblemwithworkforceconstraintscannotbecom-paredthateasilytothetimetablingproblemwithoperatorconstraints.Inthetimetablingproblemwithworkforceconstraintsthereisonlyonetypeofoperator,butthereareanumberofthemandtheyareinterchangeable.Inthetimetablingproblemwithoperatorortoolingconstraintstherearesev-eraldifferenttypesofoperators,butofeachtypethereisonlyone.Inanycase,boththetimetablingproblemwithoperatorortoolingconstraintsandthetimetablingproblemwithworkforceconstraintsarespecialcasesoftheprojectschedulingproblemwithworkforceconstraintsdescribedinSection4.6.Inthischapterweoftenmakeadistinctionbetweenthefeasibilityversionofaproblemanditsoptimizationversion.Inthefeasibilityversionweneedtodeterminewhetherornotafeasiblescheduleexists;intheoptimizationversionanobjectivehastobeminimized.Ifnoefficientalgorithmexistsforthefeasibilityversion,thennoefficientalgorithmexistsfortheoptimizationversioneither.Throughoutthischapterweassumethatalldataareintegerandthatpreemptionsarenotallowed.9.2ReservationswithoutSlackInthissectionweconsiderthefollowingreservationmodel.Therearemre-sourcesinparallelandnactivities.Activityjhasareleasedaterj,aduedatedj,andaweightwj.Asstatedbefore,alldataareinteger.Thefactthatthereisnoslackbetweenreleasedateandduedateimpliesthatpj=dj−rj.
2109IntervalScheduling,Reservations,andTimetablingIfwedecidetodoactivityj,thenithastobedonewithinthespecifiedtimeframe.However,itmaybethecasethatactivityjcannotbedonebyjustanyoneofthemresources;itmayhavetobedonebyaresourcethatbelongstoaspecificsubsetMjofthemresources.Whenallactivitieshaveequalweights,theobjectiveistomaximizethenumberofactivitiesdone.Incontrast,whentheactivitieshavedifferentweights,theobjectiveistomaximizetheweightednumberofactivitiesscheduled.Aweightisoftenequivalenttoaprofitthatismadebydoingtheactivity.Inamoregeneralmodeltheweightofactivityjmayalsodependontheresourcetowhichitisassigned,i.e.,theweightiswij(i.e.,theprofitdependsontheactivityaswellasontheresourcetowhichtheactivityisassigned).Example9.2.1(ACarRentalAgency).Consideracarrentalagencywithfourtypesofcars:subcompact,midsize,fullsizeandsport-utility.Ofeachtypethereareafixednumberavailable.Whencustomerjcallstomakeareservationforpjdays,hemay,forexample,requestacarofeitheroneoftwotypesandwillacceptthepricequotedbytheagencyforeithertype.ThesetMjforsuchacustomerincludesallcarsbelongingtothetwotypes.Theprofitmadebytheagencyforacaroftypeiisπidollarsperday.So,theweightofthisparticularreservationiswij=πipj.However,ifcustomerjspecificallyrequestsasubcompactandallsubcom-pactshavebeenrentedout,theagencymaydecidetogivehimamidsizeforthepriceofasubcompactinordernottolosehimasacustomer.ThesetMjincludessubcompactsaswellasmidsizes(eventhoughcustomerjrequestedasubcompact),buttheagency’sdailyprofitisafunctionofthecaraswellasofthecustomer,i.e.,πijdollarsperday,sincetheagencygiveshimalargercaratalowerprice.Theweightiswij=πijpj.Mostreservationproblemscanbeformulatedasintegerprograms.Timeisdividedinperiodsorslotsofunitlength.Ifthenumberofslotsisfixed,sayH,thentheproblemisreferredtoasanH-slotproblem.Assume,forthetimebeing,thattheactivitydurationsareequalto1andletJtdenotethesetofactivitiesthatneedaresourceinslott,i.e.,duringperiod[t−1,t].Ifxijdenotesabinaryvariablethatassumesthevalue1ifactivityjisassignedtoresourceiand0otherwise,thenthefollowingconstraintshavetobesatisfied:mi=1xij≤1j=1,…,nj∈Jtxij≤1i=1,…,n,t=1,…,H.Thefirstsetofconstraintsensuresthateveryactivityisassignedtoatmostoneresourceandthesecondsetensuresthataresourceisnotassignedtomorethanoneactivityinanygivenslot.Theeasiestversionofthereservationproblemisafeasibilityproblem:doesthereexistanassignmentofactivitiestoresourceswitheveryactivitybeing
9.2ReservationswithoutSlack211assignedtoaresource?Aslightlyharderversionofthisfeasibilityproblemwouldbethefollowing:doesthereexistanassignmentofactivitiestoresourceswithactivityjbeingassignedtoaresourcebelongingtoagivensubsetMj?Itturnsoutthatthisproblemisstillrelativelyeasyandthereforeleftasanexercise.Intheoptimizationversionofthereservationproblemtheobjectiveistomaximizethetotalprofitmi=1nj=1wijxij,wheretheweightwijisequivalenttoaprofitassociatedwithassigningactivityjtoresourcei.Somespecialcasesofthisoptimizationproblemcanactuallybesolvedinpolynomialtime.Forexample,consideragainthecasewithallnactivitieshavingadurationequalto1,i.e.,pj=1forallj,andassumearbitraryresourcesubsetsMjandarbitraryweightswij.Eachtimeslotcanbeconsideredasaseparatesubproblemthatcanbesolvedasanindependentassignmentproblem(seeAppendixA).Anotherversionofthereservationmodelthatallowsforanefficientsolu-tionassumesarbitrarydurations,identicalweights(i.e.,wij=1foralliandj),andeachsetMjconsistingofallmresources(i.e.,themresourcesareidentical).Thedurations,thestartingtimes(releasedates)andthecomple-tiontimes(duedates)arearbitraryintegersandtheobjectiveistomaximizethenumberofactivitiesassigned.Thisproblemcannotbedecomposedintoanumberofindependentsubproblems(oneforeachtimeslot),sincethedura-tionsofthedifferentactivitiesmayoverlap.However,itcanbeshownthatthefollowingrelativelysimplealgorithmmaximizesthetotalnumberofactivities.Inthisalgorithmtheactivitiesareorderedinincreasingorderoftheirreleasedates,i.e.,r1≤r2≤···≤rn.SetJdenotesthesetofactivitiesalreadyselected.Algorithm9.2.2(MaximizingNumberofActivitiesAssigned).Step1.SetJ=∅andj=1.Step2.Ifaresourceisavailableattimerj,thenassignactivityjtothatresource;includeactivityjinJ,andgotoStep4.OtherwisegotoStep3.Step3.Letj∗besuchthatCj∗=maxk∈J(Ck)=maxk∈J(rk+pk).
2129IntervalScheduling,Reservations,andTimetablingIfCj=rj+pj>Cj∗,donotincludeactivityjinJandgotoStep4.Otherwise,deleteactivityj∗fromJ,assignactivityjtotheresourcefreedandincludeactivityjinJ.Step4.Ifj=n,STOP,otherwisesetj=j+1andreturntoStep2.Anotherversionofthisreservationmodelwithzeroslack,arbitrarydura-tions,equalweights,andidenticalresourcesisalsoofinterest.Assumethereareanunlimitednumberofidenticalresourcesinparallelandallactivitieshavetobeassigned.However,theassignmentmustbedoneinsuchawaythataminimumnumberofresourcesisused.Thisproblemis,inasense,adualoftheproblemdiscussedbefore.Itturnsoutthatminimizingthenumberofresourceswhenallactivitieshavetobedoneisalsoaneasyproblem.Itcanbesolvedasfollows.Again,theactivitiesareorderedinincreasingorderoftheirreleasedates,i.e.,r1≤r2≤···≤rn.First,activity1isassignedtoresource1.Thealgorithmthenproceedswithassigningtheactivities,onebyone,totheresources.Supposethatthefirstj−1activitieshavebeenassignedtoresources1,2,…,i.Someoftheseactivitiesmayhavebeenassignedtothesameresource.Soi≤j−1.Thealgorithmthentakesthenextactivityfromthelist,activityj,andtriestoassignittoaresourcethatalreadyhasbeenutilizedbefore.Ifthisisnotpossible,i.e.,resources1,…,iareallbusyattimerj,thenthealgorithmassignsactivityjtoresourcei+1.Thenumberofresourcesutilizedafteractivitynhasbeenassignedistheminimumnumberofresourcesrequired.Thislastproblemturnsouttobeaspecialcaseofawell-knownnodecol-oringproblemingraphtheory.Considernnodesandletnodejcorrespondtoactivityj.Ifthereisan(undirected)arc(j,k)connectingnodesjandk,thentheprocessingofactivitiesjandkoverlapintimeandnodesjandkcannotbegiventhesamecolor.Ifthegraphcanbecoloredwithm(orless)colors,thenafeasiblescheduleexistswithmresources.Thisnodecoloringproblem,whichisafeasibilityproblemthatisNP-hard,ismoregeneralthanthereservationprobleminwhichthenumberofresourcesusedisminimized.ThisnodecoloringproblemisactuallyequivalenttothetimetablingproblemwithoperatorortoolingconstraintsdescribedinSection9.5.Thenodecol-oringproblemestablishesthelinksbetweenintervalscheduling,reservations,andtimetabling.9.3ReservationswithSlackIntheprevioussectionweassumedthattherewasnoslackbetweenthereleasedateandtheduedateofeachactivity,i.e.,
9.3ReservationswithSlack213pj=dj−rj.Amoregeneralversionofthereservationmodelallowsforslackinthetimewindowspecified,i.e.,pj≤dj−rj.Weagainconsiderfirstthespecialcasewhereallreleasedatesandduedatesareintegerandallprocessingtimesareequaltoone.TheweightsofallactivitiesareidenticalandallMjsetsconsistofallmresources.Thiscaseistrivialsinceaschedulecanbeconstructedprogressivelyintimeandthemaximumnumberofactivitiescanbeassigned.Themoregeneralproblemwithnon-identicaldurationsdoesnothaveaneasysolution.MaximizingtheweightednumberofactivitiesassignedisNP-hard,soitisunlikelythatthereexistsanefficientalgorithmthatwouldguaranteeanoptimalsolution.Wehavetorelyonheuristics.Thefollowingheuristicisbasicallyacompositedispatchingruleasde-scribedinAppendixC.Itrequires,asafirststep,thecomputationofanum-berofstatistics.Letνitdenotethenumberofactivitiesthatmaybeassignedtoresourceiduringinterval[t−1,t].Thisfactorthuscorrespondstoapo-tentialutilizationofresourceiintimeslott.Thehigherthisnumberis,themoreflexibleresourceiisinthistimeslot.Asecondfactoristhenumberofresourcestowhichactivityjcanbeassigned,i.e.,thenumberofresourcesinsetMj,whichisdenotedby|Mj|.Thelargerthisnumber,themoreflexibleactivityjis.DefineforactivityjapriorityindexIjthatisafunctionofwj/pjand|Mj|,i.e.,Ij=f(wj/pj,|Mj|).Thehigherwj/pjandthesmaller|Mj|,thelowertheindex.Theactivitiescannowbeorderedinincreasingorderoftheirindices,i.e.,I1≤I2≤···≤In.Thealgorithmtakestheactivitywiththelowestindexamongtheremainingactivitiesandattemptstoassignittooneoftheresources,startingwiththeresourcewiththeleastflexibletimeintervals.Iftheactivityneedsaresourceovertheperiod[t,t+pj],thentheselectionofresourceidependsonafunctionofthefactorsνi,t+1,…,νi,t+pj,i.e.,g(νi,t+1,…,νi,t+pj).Examplesofsuchfunctionsare:g(νi,t+1,…,νi,t+pj)=pjl=1νi,t+l/pj;g(νi,t+1,…,νi,t+pj)=max(νi,t+1,…,νi,t+pj).Theheuristicattemptstoassigntheactivitytoaresourceinaperiodthathasthelowestpossibleg(νi,t+1,…,νi,t+pj)value.Thisone-passheuristiccanbesummarizedasfollows.
2149IntervalScheduling,Reservations,andTimetablingAlgorithm9.3.1(MaximizingWeightedNumberofActivities).Step1.Setj=1Step2.Takeactivityjandselect,amongtheresourcesandtimeslotsavailable,theresourceandtimeslotswiththelowestg(νi,t+1,…,νi,t+pj)rank.Discardactivityjifitcannotbeassignedtoanymachineatanytime.Step3.Ifj=nSTOP,otherwisesetj=j+1andreturntoStep2.Thenextexampleillustratesthisheuristic.Example9.3.2(MaximizingWeightedNumberofActivities).Con-sidersevenactivitiesandthreeresources.activities1234567pj31094653wj2332123rj5023245dj12102015181914Mj{1,3}{1,2}{1,2,3}{2,3}{1}{1}{1,2}ConsidertheindexfunctionIj=f(wj/pj,Mj)=|Mj|wj/pj.Theindicesfortheactivitiescanbecomputedandaretabulatedbelow.activities1234567Ij36.679462.52Thefactorsνitaretabulatedbelow.slott012345678910111213141516171819ν1t11334666665544333321ν2t11233444443333211111ν3t00122333333322211111Applyingthealgorithmusingthefunctiong(νi,t+1,…,νi,t+pj)=pjl=1νi,t+l/pjyieldsthescheduledepictedinFigure9.1.
9.4TimetablingwithWorkforceConstraints215Machine 1Machine 2Machine 30102015t1245756Fig.9.1.ScheduleinExample9.3.2ActivityResourcePeriod7211-146114-19135-84311-15512-8220-10Itturnsoutthatactivity3(thelastactivity)doesnotfitintotheschedule.However,intheoptimalscheduleallactivitiesareassigned(activity7startswithresource1attime10andactivity3startswithresource2attime11).Sotheheuristicyieldsinthiscaseasuboptimalsolution.Thefunctionf(wj/pj,|Mj|)=|Mj|2wj/pj,yieldsthesamescheduleastheonedescribedabove(althoughthesequenceinwhichtheactivitiesareputontheresourcesisslightlydifferent,thefinalresultisstillthesame).Thefunctionf(wj/pj,|Mj|)=|Mj|wj/pjyieldsaschedulewithactivity3assigned.However,nowactivity5endsupunassigned.Notingthatw5=1andw3=3,thislastscheduleisactuallybetterthanthepreviousone,butstillnotoptimal.9.4TimetablingwithWorkforceConstraintsConsidernowaninfinitenumberofidenticalresourcesinparallel.Therearenactivitiesandallactivitieshavetobedone.Activityjcanbedonebyoron
2169IntervalScheduling,Reservations,andTimetablinganyoneoftheresources,butoncetheactivityhasstartedithastoproceedwithoutinterruptionuntilitiscompleted.ThereisaworkforcethatconsistsofasinglepoolofW1identicaloperators.InordertodoactivityjitisnecessarytohaveW1joperatorsathand.IfthesumoftherequirementsofactivitiesjandkislargerthanW1,i.e.,W1j+W1k>W1,thenactivitiesjandkcannotbedoneatthesametime.Actually,thesumoftherequirementsofanysetofactivitiesthataredoneatanygivenpointintimemaynotexceedW1.ItisclearthatthisproblemisaspecialcaseoftheworkforceconstrainedprojectschedulingproblemdescribedinSection4.6.Suchamodelwithworkforceconstraintscanbeusedforworkforceschedul-ingapplications.However,workforceschedulingproblemsingeneraltendtobemorecomplicatedandwillbediscussedinmoredetailinChapter13.Example9.4.1(ProjectManagementintheConstructionIndustry).Acontractorhastocompletenactivities.ThedurationofactivityjispjanditrequiresacrewofsizeW1j.Theactivitiesarenotsubjecttoprecedenceconstraints.ThecontractorhasW1workersathisdisposalandhisobjectiveistocompleteallnactivitiesinminimumtime.Considernowthefollowingspecialcaseoftheworkforceconstrainedtimetablingproblemwiththenumberofavailableresourcesbeingunlimited.Allactivitieshavenowthesameduration.Theactivitiesarenotsubjecttoprecedenceconstraintsbutaresubjecttoworkforceconstraintsandtheobjec-tiveistominimizethemakespan.Thisproblemmayberegardedasadiscretecounterpartofafamouscombinatorialproblemknownasthebinpackingproblem.InthebinpackingproblemeachbinhascapacityW1andactivityjisequivalenttoanitemofsizeW1j.Eachbincorrespondstoonetimeslotandtheitemspackedinonebincorrespondtotheactivitiesdoneinthattimeslot.Theobjectiveistopackalltheitemsinaminimumnumberofbins.Thisproblemhasmanyapplicationsinpractice.Example9.4.2(ExamScheduling).Alltheexamsinacommunitycollegehavethesameduration.TheexamshavetobeheldinagymwithW1seats.TheenrollmentincoursejisW1jandallW1jstudentshavetotaketheexamatthesametime.Thegoalistodevelopatimetablethatschedulesallnexamsinminimumtime.ThisworkforceconstrainedschedulingproblemwithallactivitieshavingthesamedurationisknowntobeNP-hard(evenintheabsenceofprece-denceconstraints).However,anumberofheuristicshavebeendevelopedthatperformreasonablywell.TheFirstFit(FF)heuristicfirstorderstheactivities(items)inanarbi-traryway.Theslots(bins)arenumbered1,2,3,…Theprocedurestartsatthebeginningoftheactivitylistandcheckswhethertheactivityfitsinslot1.Ifitfits,itisinsertedthere.Otherwise,theprocedurecheckswhetheritfitsinslot2,andsoon.Ithasbeenshownthatforanyinstanceofthisproblem
9.4TimetablingwithWorkforceConstraints217Cmax(FF)≤1710Cmax(OPT)+2,whereCmax(FF)denotesthemakespanundertheFFruleandCmax(OPT)denotesthemakespanundera(possiblyunknown)optimalrule.ItiseasytofindinstancesforwhichCmax(FF)Cmax(OPT)=53.Example9.4.3(ApplicationoftheFFHeuristic).LetW1=2100.Thereare18activities.activities1,…,67,…,1213,…,18W1j3017011051UndertheoptimalschedulethemakespanCmaxis6andundertheFFschedulethemakespanisequalto10.Theoptimalscheduleassignstoeachoneofthesixslotsthreeactivities:oneof301,oneof701andoneof1051.TheFFruleassignssixactivitiesof301toslot1.Toeachoneofthenextthreeslotsitassignstwoactivitiesof701.Toeachoneofthelastsixslotsitassignsasingleactivityof1051.ThisexampleshowsthatanFFschedulemaybefarfromoptimalwhentheactivitiesinitiallyarelistedinahaphazardway.Iftheactivitiesareor-deredinitiallyinacleverwaytheFirstFitheuristicmayperformbetter.Theworstcaseperformanceofthenextheuristic,thatisbasedonthisidea,issignificantlybetter.TheFirstFitDecreasing(FFD)heuristicfirstorderstheactivitiesinde-creasingorderofW1j.Theslotsareagainnumbered1,2,3,andsoon.Theprocedurestartsatthebeginningoftheactivitylistandcheckswhethertheactivityfitsinslot1.Ifitfits,itisinserted.Otherwise,theprocedurecheckswhethertheactivityfitsinslot2,andsoon.IthasbeenshownthatforanyinstanceoftheproblemCmax(FFD)≤119Cmax(OPT)+4,whereCmax(FFD)denotesthemakespanundertheFFDrule.ThereareinstancesforwhichCmax(FFD)Cmax(OPT)=119.Thefollowingexampleshowshowthisworstcaseboundcanbeattained.Example9.4.4(ApplicationoftheFFDHeuristic).LetW1=1000.activities1,…,67,…,1213,…,1819,…,30W1j501252251248
2189IntervalScheduling,Reservations,andTimetablingUndertheoptimalschedulethemakespanCmaxis9andundertheFFDschedulethemakespanis11.Theoptimalscheduleassignstoeachoneofthefirstsixslotsthreeactivities:oneof501,oneof251andoneof248.Toeachoneoftheremainingthreeslotsitassignsfouractivities:twoof252andtwoof248.TheFFDruleassignstoeachoneofthefirstsixslotsoneactivityof501andoneactivityof252.Tothenexttwoslotsitassignsthreeactivitiesof251.Toeachoneofthelastthreeslotsitassignsfouractivitiesof248.Thetwoheuristicsdescribedabovecanbeappliedwithsomeminormodi-ficationstocaseswheretheactivitieshavedifferentreleasedates.Iftheactivi-tieshaveduedatesandtheobjectiveistheminimizationofaduedaterelatedpenaltyfunction,thenadifferentheuristicisrequired.Iftheactivitieshavedeadlinesandtheobjectiveistofindafeasibleschedule,thenalsoadditionalmodificationsareneeded.9.5TimetablingwithOperatororToolingConstraintsIntheprevioussectionweconsideredmodelswithW1identicaloperators.Theoperatorswerebasicallyinterchangeable.Inwhatfollowstheoperatorsarenotidentical.Eachoperatorisunique,hashisownidentityandhisownskill.Anoperatorinthismodelmayactuallybeequivalenttoaspecificpieceofmachinery,afixture,oratool.Anactivityeitherneedsordoesnotneedanygivenoperatorortool.Eachactivityneedsforitsexecutionaspecificsetofdifferentoperatorsand/ortools.Itiseasytoseethatthisproblemisalsoaspecialcaseofthework-forceconstrainedprojectschedulingproblemdiscussedinSection4.6.Itisaprojectschedulingproblemwithworkforceconstraintsandwithoutprece-denceconstraints.TherearenowNpdifferentpoolsofoperatorsandeachpoolofoperatorsconsistsofasingleoperator,i.e.,W=1for=1,…,Np.Thedifferencebetweentimetablingwithoperatorortoolingconstraintsandtimetablingwithworkforceconstraintsissignificant.Inonesensethemodelintheprevioussectionismorerestrictive(thereisonlyonetypeofoperator,i.e.,Np=1),andinanothersenseitismoregeneral(thereareanumberofthattypeofoperatoravailable,i.e.,W1>1).Eachactivitynowrequiresoneormoredifferentoperatorsortools.Iftwoactivitiesrequirethesameoperator,thentheycannotbedoneatthesametime.Inthefeasibilityversionofthisproblem,thegoalistofindascheduleortimetablethatcompletesallnactivitieswithinthetimehorizonH.Intheoptimizationversion,theobjectiveistodoalltheactivitiesandminimizethemakespan.Throughoutthissectionweassumethatallactivitydurationsareequalto1.Eventhisspecialcasewithallactivitydurationsbeingequaldoesnothaveaneasysolution.Inwhatfollowswefirstfocusonthefeasibilityversionwhen
9.5TimetablingwithOperatororToolingConstraints219alldurationsareequalto1.Findingforthiscaseaconflict-freetimetableisstructurallyequivalenttothenodecoloringproblemdescribedattheendofSection9.2.Inthenodecoloringproblemagraphisconstructedbyrepresent-ingeachactivityasanode.Twonodesareconnectedbyanarcifthetwoactivitiesrequirethesameoperator(s).Thetwoactivities,therefore,cannotbescheduledinthesametimeslot.IfthelengthofthetimehorizonisHtimeslots,thenthequestionis:canthenodesinthegraphbecoloredwithHdifferentcolorsinsuchawaythatnotwoconnectednodesreceivethesamecolor?Thisisafeasibilityproblem.Theassociatedoptimizationproblemistodeterminetheminimumnumberofcolorsneededtocolorthenodesofthegraphinsuchawaythatnotwoconnectednodeshavethesamecolor.Thisminimumnumberofcolorsisreferredtoasthechromaticnumberofthegraphandisequivalenttothemakespaninthetimetablingproblem.Theoptimizationversionofthetimetablingproblemwithalldurationsbeingequalto1iscloselyrelatedtothezeroslackreservationproblemwitharbitrarydurationsdescribedattheendofSection9.2.Thatthisreservationproblem(withthenumberofresourcesbeingminimized)isnotequivalenttothetimetablingproblembutratheraspecialcasecanbeshownasfollows:twoactivitiesthatneedthesameoperatorinthetimetablingproblemareequivalenttotwoactivitiesthathaveanoverlappingtimeslotinthereserva-tionproblem.Iftwoactivitiesinthereservationproblemhaveanoverlappingtimeslot,thenthetwonodesareconnected.Eachcolorinthecoloringpro-cessrepresentsaresourceandminimizingthenumberofcolorsisequivalenttominimizingthenumberofresourcesinthereservationproblem.Thatthereservationproblemisaspecialcasefollowsfromthefactthatthetimeslotsrequiredbyanactivityinareservationproblemareadjacent.However,itmaynotbepossibletoorderthetoolsinthetimetablingprobleminsuchawaythatthetoolsrequiredforeachactivityareadjacenttooneanother.Itisthisadjacencypropertythatmakesthereservationproblemeasy,whilethelackofadjacencymakesthetimetablingproblemwithoperatorconstraintshard.Thereareanumberofheuristicsforthistimetablingproblemwithdura-tionsequalto1.Inthissectionwedescribeonlyonesuchprocedure.Firstsomegraphtheoryterminologyisneeded:thedegreeofanodeisthenumberofarcsconnectedtoanode;inapartiallycoloredgraph,thesaturationlevelofanodeisthenumberofdifferentlycolorednodesalreadyconnectedtoit.Inthecoloringprocess,thefirstcolortobeusedislabeledColor1,thesecondColor2,andsoon.Algorithm9.5.1(GraphColoringHeuristic).Step1.Arrangethenodesindecreasingorderoftheirdegree.Step2.ColoranodeofmaximaldegreewithColor1.
2209IntervalScheduling,Reservations,andTimetablingStep3.Chooseanuncolorednodewithmaximalsaturationlevel.Ifthereisatie,chooseanyoneofthenodeswithmaximaldegreeintheuncoloredsubgraph.Step4.Colortheselectednodewiththecolorwiththelowestpossiblenumber.Step5.Ifallnodesarecolored,STOP.OtherwisegotoStep3.Example9.5.2(ApplicationoftheGraphColoringHeuristic).Gary,Hamilton,IzakandRehaareuniversityprofessorsattendinganationalcon-ference.Duringthisconferencesevenonehourmeetingshavetobescheduledinsuchawaythateachoneofthefourprofessorscanbepresentatallthemeetingshehastoattend.Thegoalistoscheduleallsevenmeetingsinasingleafternoonbetween2p.m.and6p.m.meetings1234567Gary1001101Hamilton1110000Izak0010110Reha1011100Thisproblemcanbetransformedintoatimetablingproblemwithoperatorconstraintsbyassumingthatthesevenmeetingsareactivitiesandthefourprofessorsareoperators.Considerthefollowingsetofdata.activities1234567operator11001101operator21110000operator30010110operator41011100Iftheactivitiesareregardedasnodes,thentheirdegreescanbecomputed(seeFigure9.2).activities(nodes)1234567degree5254523
9.5TimetablingwithOperatororToolingConstraints2211762543Fig.9.2.GraphinExample9.5.2Basedonthedegrees,activity5maybecoloredfirst,saywiththecolorred(Color1).Thesaturationlevelsofallnodesconnectedtonode5,i.e.,nodes1,3,4,6,7,areequalto1.Ofthesenodes,nodes1and3havethehighestdegrees.Colornode3blue(Color2).Thesaturationlevelsandthedegreesintheuncoloredsubgrapharepresentedinthetablebelow.activities(nodes)1234567saturationlevel21-2-21degree31-2-02Basedonthesenumbersnode1isselectedasthenodetobecolorednext,sayyellow(Color3).Node4isselectedafterthatandcoloredgreen(Color4).Node7followsandiscoloredwiththecolorthathasthelowestnumber,Color2(blue).Node6iscoloredlastandcoloredyellow.Sincefourcolorswereneededtocolorthegraph,themakespanofthecorrespondingscheduleisequalto4.Itcaneasilybeseenthatthisscheduleisoptimal.Bothoperators1and4areneededfor4activities.Toseewhyitmakessensetoscheduletheactivitywiththehighestdegreefirst,considerschedulingtheactivitywiththelowestdegreefirst.Activities2,6,and7thenhavetobedoneinthesametimeslot.However,activities4,1,3,and5arescheduledafterwardsinfourdifferenttimeslotsandthemakespanis5.Ifthehighdegreeactivitiesarenotscheduledearlyon,theyoftenenduprequiringnewcolorsattheendoftheprocess.Thenextexampleillustratestherelationshipbetweenthereservationprob-lemandthetimetablingproblem.Example9.5.3(TimetablingComparedtoReservations).Considerthefollowingtimetablingproblem.
2229IntervalScheduling,Reservations,andTimetablingactivities1234567pj1111111operator11011101operator21110000operator30010110operator41011100Notethattheonlydifferencebetweenthisexampleandthepreviousoneisthatactivity3nowneedsallfouroperators.Theoperatorscanbetransformedintotimeslotsasfollows.Operators3and4areequivalenttotimeslots1and2andoperators1and2areequivalenttotimeslots3and4.Thetimeslotsrequiredbyeachactivityarenowcontigious(i.e.,adjacent)andtheproblemisequivalenttoareservationproblem.Considernowamoregeneraltimetablingmodelwithallactivitiesagainhavingduration1.Thereareanumberoffeasibleslots.However,thereisanaversioncostcjtforassigningactivityjtoslott.Thereisalsoaproximitycostforschedulingtwoconflictingactivities(thatrequirethesameoperator)tooclosetooneanother.Thepenaltyforschedulingtwoconflictingactivitiesτslotsapartisψ(τ),whereψ(τ)isdecreasinginτ.Theobjectivefunctionis,foranyschedule,thesumofthesetwocostsforeveryoccurrence.Thefollowingmulti-passheuristicisapplicabletothismoregeneraltimetablingproblem.Algorithm9.5.4(MinimizingTimetablingCosts).Step1.Takeactivityjfromthesetofactivitiesnotyetscheduled.Step2.Findallfeasibleslotswhereactivityjcanbeassigned,i.e.,wherethereisnooperatorconflict.IfnosuchslotisfoundgotoStep4.Step3.Foreachfeasibleslot,computetheincreaseinthecostfunction(aversionaswellasproximitycosts);Assignactivityjtotheslotwiththelowestcost.GotoStep1.Step4.Activityjconflictsineverypossibleslotwithotheractivities.Findtheslotsinwhichactivityjcanbescheduledbyreschedulingallac-tivitiesthatconflictwithj.IftherearenosuchslotsgotoStep6.OtherwisegotoStep5.
9.5TimetablingwithOperatororToolingConstraints223Step5.Foreachslotcalculatethecostofreschedulingallconflictingactivities.Assignactivityjtothatslotwiththelowestreschedulingcost.Step6.Iftherearenoslotsforwhichconflictingactivitiescanberescheduledwith-outconflict,countforeachslotthenumberofactivitiesthatcannotberescheduled.Assignactivityjtothatslotwiththeleastnumberofsuchconflicts.Rescheduleasmanyconflictsaspossibleandbumptheothersbackonthelistofunscheduledactivities.IfthenumberoftimesactivitykisbumpedbythesameactivityjreachesN,thenactivitykisdroppedandconsideredunschedulable.Thelaststepofthealgorithmcanbeviewedasaboundedbacktrackingmechanismthatallowsittoreconsiderearlierdecisions.Ifapairofactivitiesisdifficulttoschedule,thentheymostlikelywillbumpeachotherrelativelyearlyinthebacktrackingprocess.Thenextexampleillustratesthemannerinwhichthebacktrackingmechanismworkswhenthealgorithmisappliedtothemorespecificmodeldescribedearlierinthissection.Example9.5.5(MinimizingTimetablingCosts).ConsidertheinstancediscussedinExample9.5.2.Assumethatthenumberoffeasibleslotsis4(thisimpliesthatthealgorithmwilleitherfindtheoptimalscheduleoritwillconcludethatthereisnofeasibleschedule).Assumethatallaversioncostsandproximitycostsarezero.SinceAlgorithm9.5.4doesnotspecifytheorderinwhichtheunsched-uledactivitiesaretaken,weassumeherethattheactivitiesaregoingtobeconsideredintheorder2,6,7,4,1,3,5.GoingthroughSteps1,2,and3anumberoftimesresultsinactivities2,6,and7beingdoneinslot[0,1],activity4inslot[1,2],activity1in[2,3],andactivity3in[3,4].However,whenthealgorithmattemptstoinsertthelastactivity,activity5,inoneofthefourslots,thenthereareconflictsineachoneofthem.Ifactivity5isputintheslotofactivity4,thenactivity4hastoberescheduled,butactivity4cannotberescheduledinanyoftheotherthreeslots.Thesamethinghappensifactivity5isassignedtothethirdorfourthslot.Sothefirstslotremainstobechecked.Ifactivity5isinsertedinthefirstslot,thenactivities6and7areinconflictandhavetoberescheduled.Activity6canbescheduledtogetherwithactivity4inthesecondslotandactivity7canbescheduledtogetherwithactivity3inthefourthslot.Sotheoptimalschedulehasbeenobtained.
2249IntervalScheduling,Reservations,andTimetabling9.6AssigningClassestoRoomsatU.C.BerkeleyTheUniversityofCaliforniaatBerkeleyenrollsabout30,000studentsinover80academicdepartments.Eachsemester,alldepartmentsprovidetheschedulingofficeanestimatedenrollment,arequestedmeetingtime,andspe-cialrequirements(e.g.,withregardtoaudiovisualequipment)foreachsectionofeachcourse.Theschedulingofficemustassign4000classestoabout250classrooms.Theofficeconsistsofthreeschedulers(oneforeachacademicunit)andonesupervisor.Theassignmenthastotakeanumberofobjectivesintoconsideration.Aroomwithfewerseatsthanstudentsisundesirable,asisaroomthatismuchtoolarge.Inaddition,somecoursesrequirespecialequipment.Thelocationoftheroomisalsoimportant.Fromaprofessor’spointofview,itisnicetohavearoomthatisclosetohisorheroffice.Fromastudents’pointofviewitisconvenienttohaveconsecutiveclassesclosetogether.Itisnoteasytostateaformalobjectiveforthisoptimizationproblem,sincethereareoftennoclearpriorities.Forexample,ifthereisnoroomtoaccomodatebothChemistry201andRussian101atthesametime,thenitisnoteasytomakeachoicebasedonsomegeneralprinciple.Fortunately,somepolicyguidelineshadbeenestablishedbyacampuscommittee.Thepolicyguidelinesarebasedonstandardtimepatternsforofferingcourses.Thenine-hourday,startingat8a.m.,ispartitionedintonineonehourblocksand,atthesametime,alsointo6oneandone-half-hourblocks.Classesmaybescheduledonlyforwholeblocks.Certaintimeblocksaredefinedasprimetimeanddepartmentsmaynotrequestmorethan60%oftheircoursesduringprimetime.Standardcourseshavepriorityovernonstandardcourses.Thesepoliciesprovidesomemeansofdecidingwhichcoursesshouldnotbeassignedduringoverloadedtimeblocks.However,theystilldonotprovideawatertightmethodforresolvingconflictsbetweendepartments.Thisclassroomassignmentproblemcanbeformulatedasalarge0−1integerprogrammingproblem.Theobjectivefunctionofthisintegerprogramisrathercomplicatedandcontainsmanyterms.First,thereisapenaltyas-sociatedfornotassigningaclassatall.Bymakingthispenaltylargerelativetotheothertermsintheobjective,thetotalnumberofunassignedclassesisminimized.Thecosttermsintheobjectiveassociatedwiththeassignmentvariablesaccountfordistances,overutilizedfacilities,andemptyseats.Sincetheintegerprogramishuge(approximately500,000variablesand30,000constraints),itissolvedheuristicallyeventhoughtheproblemdoesnothavetobesolvedinrealtime.Theheuristicworksinasequentialmannerandisbasedontheprincipleofalwayssolvingthehardestremainingsubproblemnext.Somenotationisneededinordertodescribetheheuristic.LetJdenotethesetofallclassestobescheduled.LettdenoteatimeslotandletJtdenotethesetofallclassesfortimeslott.LetMdenotethesetofallclassroomsandletMjdenotethesetofallclassroomsthatcanaccomodateclassj.
9.6AssigningClassestoRoomsatU.C.Berkeley225Theheuristiccanbesummarizedinthefollowingfoursteps.Algorithm9.6.1(RoomAssignmentHeuristic).Step1.(SelectTimeSlot)Select,amongtimeslotsnotyetconsidered,slottwiththesmallestsup-ply/demandratio.Step2.(GreedyAlgorithm)RankallclassesjinJtindecreasingorderofclasssize.Goinasinglepassthroughthelistofclasses,andassignclassjtothe(stillvacant)roominMjwithlowestcost.Step3.(ImprovementPhase)RankallclassesjinJtindecreasingorderofcurrentcost.Goinasinglepassthroughthelistofclassesanddothefollowing:Ifclassjisnotassigned,findallfeasibleinterchangesinwhichclassjmovesintoanoccupiedroomdisplacingtheassignedclasskintoavacantroom;ifthissetisnotempty,maketheinterchangewithmaximumcostreduction.Ifclassjisassigned,findthesetoffeasibleassignmentinterchangesforclassjthatreducetotalcost;ifthissetisnotempty,applytheinterchangewiththemaximumcostreduction.Step4.(StoppingCriterion)IfStep3hasresultedinareductionofthetotalcostreturntoStep3;oth-erwisedeletefromtheunscheduledlistallclassesscheduledforthecurrenttimeslot.IfalltimeslotshavebeenconsideredSTOP,otherwisegotoStep1.Theheuristichasproventobeaveryfastmethodforgeneratingnear-optimalsolutions.Combiningtheruleof“selectingthehardestsubproblemnext”withadynamicrecalculationofthecostsofwastedresourcesseemsveryeffective.Thedecisionsupportsystemisdesignedsothatitiseasytousein-teractivelyanditisflexibleenoughtoaccomodatefuturepolicymodificationswithoutextensivereprogramming.Thesystemisusedinthefollowingmanner.Approximatelysixmonthsbeforethestartofthesemester,departmentssubmitroomrequestformsthatlistallclassesscheduledforthesemester.Withinacoupleofweeks,aprelim-inaryscheduleisgenerated,showingthoseclassesthatcouldnotbeassignedtorooms.Departmentsthensubmitrevisedrequestsandnegotiatewiththeschedulingofficeaboutpossiblepre-assignments.Thesystemisthenrunagain,withtheunchangedstandardlecturesthathadalreadybeenassignedflaggedaspre-assigned.Theresultingsetofassignmentsisthenpublishedintimeforpre-enrollment.Usingthesystem,theschedulingofficeisabletocompleteitspartofthecycleseveralweeksearlierthanwiththeoriginalmanualprocedure.
2269IntervalScheduling,Reservations,andTimetablingThesystemhasbeenusedforanumberofyears.Anumberoffactorscontributedtothesuccessofthesystem.Themostimportantoneisaflexi-bleuserinterface.Whileanoptimizationmodelisbeingused,itsbehavioriseasilyalteredtoexploredifferenttrade-offstrategies.Furthermore,thesys-temisdesignedsothatpartialsolutions,intheformofeasilygeneratedpre-assignments,canbeincorporated,whileallowingtheheuristictogenerateasolutionfortheremainingproblem.Specialneedsthatwerenotanticipatedwhenthemodelwasdesignedcanbeaccomodatedandtheschedulerscanevaluatetheirownheuristics.Thisclassroomassignmentproblemisverysimilartothereservationprob-lemswithslackandwithoutslackdescribedinSections9.2and9.3.Themeet-ingroomsaretheresourcesandactivityjcanonlybeassignedtoaresourcebelongingtothesubsetMj.However,intheclassroomassignmentproblemallactivitieshavetobescheduled.Themulti-passheuristicimplementedhereisclearlymoresophisticatedthantheone-passheuristicdescribedinAlgorithm9.3.1.9.7DiscussionEventhoughthefoursectionsofthischapterdealwithfourdifferenttypesofschedulingandtimetablingproblems,itisnothardtoimaginerealworldschedulingproblemsthathaveallthefeaturesdiscussedinthischapter,i.e.,releasedatesandduedates(withorwithoutslack),workforceconstraintsaswellasoperatorconstraints.Theobjectivemayalsobeacombinationoftheminimizationofthemakespanandthemaximizationofthe(weighted)numberofactivitiesdone.Theseparateanalysesofthesefourdifferentaspectsgiveanindicationofhowhardrealworldproblemscanbe.Theintervalschedulingmodelsandreservationmodelsdiscussedinthischapterarerelativelysimple.Theyonlygiveaflavorofthethinkingbehindtheseproblems.Acompanythathastodealwiththesetypesofproblemstypicallyreliesonmodelsthataresignificantlymorecomplicated.Firstofall,themodelshavetobedynamicratherthanstatic.Callsforreservationscomeincontinuouslyandthedecisionmakingprocessembeddedinthecom-pany’ssystemsdependsheavilyonforecastsoffuturedemands.Second,thereservationsystemsdependontheexistingpricingstructureandthepricingstructuredependsonthecurrentoccupancyaswellasonforecastdemand.Asignificantamountofresearchhasbeendonerecentlyonreservationmodelsthatincludeapricingmechanism.Thesemodelsarebeyondthescopeofthisbook.Anaturalgeneralizationoftimetablingproblemswithoperatorconstraintsandtimetablingproblemswithworkforceconstraintsisthefollowing:Supposethereareanumberofdifferenttypesofoperators,sayNp.AlimitednumberWofoperatorsoftype,=1,…,Np,areavailable.DoingactivityjrequiresWjoperatorsoftype.Theactivitieshavetobescheduledsubjecttothe
Exercises227operatoravailabilityconstraints.TheFFandFFDrulescanbeadaptedtothissituationwithmultipletypesofoperators.ThisproblemisbasicallyequivalenttotheproblemdiscussedinSection4.6withoutprecedenceconstraints.Inthischapterwehavenotconsideredanypreemptions.Whenthedu-rationsoftheactivitiesarenotequalto1,preemptionsmayimprovetheperformancemeasures.However,heuristicsforpreeemptivemodelsmaybeverydifferentfromthosefornonpreemptivemodels.Exercises9.1.Considerthefollowingreservationproblemwith10activitiesandzeroslack.Therearethreeidenticalresourcesinparallel.activities12345678910pj6142336213rj2752104800dj889443101013(a)ApplyAlgorithm9.2.1tofindtheschedulewiththemaximumnumberofactivitiesdone.(b)Findtheschedulethatmaximizesthetotalamountofprocessing(i.e.,thesumofthedurationsoftheactivitiesdone).(c)Whatistheminimumnumberofresourcesneededtosatisfythetotaldemand?9.2.Consideracarrentalagencywherethefollowing10reservationshavebeenmade.reservations12345678910pj3142311333rj1301002342dj4443313675Determinetheminimumnumberofcarsneededtosatisfythedemand.9.3.Considerthefollowinginstancewitheightactivitiesandthreeresources.activities12345678pj431094653wj32332123rj85023245dj1212102015181914Mj{2}{1,3}{1,2}{1,2,3}{2,3}{1}{1}{1,2}
2289IntervalScheduling,Reservations,andTimetabling(a)SelectappropriateindexfunctionsIjandg(νi,t+1,…,νi,t+pj)andap-plyAlgorithm9.3.1.(b)Isthesolutionobtainedoptimal?Ifnot,canyoumodifythealgorithmtoobtainanoptimalsolution?9.4.Considerahotelwithtwotypesofrooms:suitesandregularrooms.Therearen1suitesandn2regularrooms.Ifsomeonewantsasuite,thenthehotelmakesaprofitofw1dollarspernight.Ifsomeonewantsaregularroom,thehotelmakesaprofitofw2dollarspernight(w2t.Thefollowingintegerprogramcanbeformulatedtomaximizethetotalprofit.maximize(j,t)∈Aπjtxjtsubjecttot:(j,t)∈Axjt≤1forj=1,…,n(j,t)∈Axjtbjtv=1forv=1,…,Hxjt∈{0,1}for(j,t)∈AThisintegerprogramtakesintoaccountthefactthatthereareshowsofdifferentdurations.However,theformulationaboveisstilltoosimpletobe
10.6SchedulingaCollegeBasketballConference247ofanypracticaluse.Oneimportantissueintelevisionbroadcastingrevolvesaroundso-calledlead-ineffects.Theseeffectsmayhaveaconsiderableimpactontheratings(andtheprofits)oftheshows.Ifaverypopularshowisfollowedbyanewshowforwhichitwouldbehardtoforecasttheratings,thenthehighratingsofthepopularshowmayhaveaspill-overeffectonthenewshow;theratingsofthenewshowmaybeenhancedbytheratingsofthepopularshow.Incorporatinglead-ineffectsintheformulationdescribedabovecanbedoneinseveralways.Onewaycanbedescribedasfollows:let(j,t,k,u)refertoalead-inconditionthatinvolvesshowjstartinginslottandshowkstartinginslotu.LetLdenotethesetofallpossiblelead-inconditions.Thebinarydecisionvariableyjtkuis1ifinaschedulethelead-incondition(j,t,k,u)isindeedineffectand0otherwise.Letπjtkudenotetheadditionalcontributiontotheobjectivefunctionifthelead-inconditionissatisfied.Theobjectivefunctionintheformulationabovehastobeexpandedwiththeterm(j,t,k,u)∈Lπjtkuyjtkuandthefollowingconstraintshavetobeadded:yjtku−xjt≤0for(j,t,k,u)∈Lyjtku−xku≤0for(j,t,k,u)∈L−yjtku+xjt+xku≤1for(j,t,k,u)∈Lyjtku∈{0,1}for(j,t,k,u)∈LThefirstsetofconstraintsensuresthatyjtkucanneverbe1whenxjtiszero.Thesecondsetofconstraintsissimilar.Thethirdsetofconstraintsensuresthatyjtkunevercanbe0whenbothxjtandxkuareequalto1.10.6SchedulingaCollegeBasketballConferenceTheAtlanticCoastConference(ACC)isagroupofnineuniversitiesinthesoutheasternUnitedStatesthatcompeteagainsteachotherinanumberofsports.Fromarevenuepointofview,themostimportantsportisbasketball.Mostoftherevenuescomefromtelevisionnetworksthatbroadcastthegamesandfromgatereceipts.Thetournamentschedulehasanimpactontherev-enuestream.Televisionnetworksneedaregularstreamofqualitygamesandspectatorswantneithertoofewnortoomanyhomegamesinanyperiod.TheACCconsistsofnineuniversities:Clemson(Clem),Duke(Duke),FloridaState(FSU),GeorgiaTech(GT),Maryland(UMD),NorthCarolina(NC),NorthCarolinaState(NCSt),Virginia(UVA)andWakeForest(Wake).Everyyear,theirbasketballteamsplayadoubleroundrobintournamentinthefirsttwomonthsoftheyear.Eachteamplayseveryotherteamtwice,onceathomeandonceaway.Usually,ateamplaystwiceaweek,oftenon
24810SchedulingandTimetablinginSportsandEntertainmentwednesdayandonsaturday(thesetwoslotsarereferredtoastheweekdayandtheweekendslot).Becausethetotalnumberofteamsisodd,therewillbeineachslotoneteamwithaBye.Ineachslottherearefourconferencegames.Theentirescheduleconsiststhereforeof18slots,whichimpliesthatthelengthofthescheduleis9weeks.Everyteamplays8slotsatHome,8Away,andhastwoByes.Therearenumerousrestrictionsintheformofpatternconstraints,gamecountconstraints,andteampairingconstraints.ThepatternsofHomegamesandAwaygamesisimportantbecauseofwearandtearontheteams,issuesofmissingclasstime,andspectatorpreferences.NoteamshouldplaymorethantwoAwaygamesconsecutively,normorethantwoHomegamesconsecutively.AByeisusuallyregardedasanAwaygame.Similarrulesapplytoweekendslots(nomorethantwoatHomeinconsecutiveweekends).Inaddition,thefirstfiveweekendsareusedforrecruitingfuturestudent-athletes,soeachteammusthaveatleasttwoHomeoroneHomeandoneByeweekendamongthefirstfive.AByeisacceptableherebecausetheopenslotcouldbeusedtoscheduleanon-conferenceHomegame.Thelastweekoftheseasonisofgreatimportancetoallteams,sonoteamcanbeAwayinbothslotsofthefinalweek.Thefinalweekendoftheseasonisthemostimportantslot,andisreservedfor”rival”pairings;thegamesDuke-UNC,Clem-GT,NCSt-Wake,andUMD-UVAareusuallyplayedonthatday.Duke-UNCisthemostcriticalpairingintheschedule.Itmustoccurinslot17andalsoinslot10.Sinceeveryteamplaystwogamesagainsteveryotherteam,theconfer-encepreferstohavethetwogamessomewhatapart.Aseparationofnineslotscanbeachievedbymirroringasingleroundrobinschedule.Becauseofcertainfixedgameassignments,itturnsoutthataperfectmirrorinthiscaseisnotpossible.However,asimilarideaisusedinordertoensurelargeseparations.AnapproachsimilartoAlgorithm10.2.2wasadoptedtocreateaschedule.However,themirroringoftheschedulewasalreadydoneinStep1ofthisalgorithmratherthanasanadditionalstepafterStep3.Step1createdHAPsoflength18.AnumberofdifferentpatternsetswerecreatedinStep1bysolvingaseriesofintegerprograms.GeneratingthetimetablesinStep2canalsobedonethroughanintegerprogramofthetypedescribedinSection10.2.Step3requiresalsoasignificantcomputationaleffort.Foreachtimetablethereare9!=362,880assignmentsofteamstopatterns.Eachoftheseischeckedforfeasibilityaspects(e.g.,arethefinalgamestherightgames)andforpreferenceas-pects(e.g.,thenumberofprimeTVslotsandthenumberofslotsthatarenotprime).TheschedulegenerationprocessisillustratedbytheflowchartinFigure10.3thatalsosummarizeswhichconstraintsareenforcedineachstep.
10.6SchedulingaCollegeBasketballConference249(A)FindFeasiblePatternsStep 1Constraints(A)Pattern(B)Game Count(C)Team PairingMirroring(D)Final SlotTV PreferencesTeam RequirementsOpponent Ordering(E)PreferencesIntangibles(B)FindPatternSetsStep 1(C)FindTimetablesStep 2(D)AssignTeamsPatternsStep 3(E)ChooseFinalSchedulePatternsPatternSetsTimetablesSchedules381717826Fig.10.3.AlgorithmFlowChartFollowingthisproceduretheofficialscheduleforthe1996-1997ACCBas-ketballTournamentwasgenerated.Thefirsthalfofthetournamentispre-sentedbelow.slot1slot2slot3slot4slot5slot6slot7slot8slot9ClemUVA-DukeFSUUMD-NCSt-WakeUNC-GTDuke-FSUGTClem-Wake-UVANCStUMD-UNCFSUDuke-NCSt-UVA-ClemGT-UNCWake-UMDGTUMD-Duke-Wake-FSUUNCUVA-NCStClemUMD-GT-UVAUNCNCSt-ClemWake-DukeFSUUNCWake-UMDUVA-NCSt-GTFSU-ClemDukeNCSt-WakeFSU-UMDUNCClem-DukeGTUVAUVA-ClemUMDFSU-UNCWakeDuke-GT-NCStWakeNCSt-UNCGTDuke-UVA-UMDClem-FSUThesecondhalfofthetournamentisasfollows.slot10slot11slot12slot13slot14slot15slot16slot17slot18ClemNCSt-UMDWake-UVADuke-FSU-UNCGTDuke-GTWake-NCStUVAFSU-Clem-UMDUNCFSUUVAUNC-GTUMD-DukeClemNCSt-WakeGTDuke-UVAFSU-UNC-UMDNCStWake-ClemUMD-Wake-NCStClem-FSUGT-UNCDukeUVAUNC-FSU-UVANCStGT-WakeUMDClem-DukeNCSt-ClemUMDDuke-UNCWake-UVA-GT-FSUUVA-FSUGTUNC-DukeClemNCSt-Wake-UMDWakeUMD-Duke-Clem-NCStUNCUVA-GTFSU
25010SchedulingandTimetablinginSportsandEntertainmentTheslotswithanevennumberareweekendslotsandtheslotswithanoddnumberareweekdayslots.Theschedulepresentedabovehasthefollowingcharacteristics:(i)theminimumdifferencebetweenrepeatinggamesis4;(ii)thenumberofstringswiththreeormoreconsecutivehomegames(con-sideringaByeasanAway)is2;(iii)thenumberofstringswiththreeormoreconsecutivehomegames(con-sideringaByeasaHome)is10;(iv)thenumberofstringswiththreeormoreconsecutiveawaygames(con-sideringaByeasaHome)is0;(v)thenumberofstringswiththreeormoreconsecutiveawaygames(con-sideringaByeasanAway)is1.Thissameschedulingproblemhasalsobeensolvedusingaconstraintpro-grammingapproach.TheapproachfollowedwasbasedontheonedescribedinSection10.3.AsfarasSteps1and2areconcerned,thecomputationalef-fortneededusingaconstraintprogrammingapproachseemstobecomparabletothecomputationaleffortneededusinganintegerprogrammingapproach.However,asfarasStep3isconcernedtheconstraintprogrammingtechniqueseemstohaveaclearedge.SincetheintegerprogrammingapproachinStep3isbasicallyequivalenttocompleteenumeration,itisnotsurprisingthatconstraintprogrammingcandobetter.10.7DiscussionInthischaptertournamentschedulingproblemshavebeendealtwiththroughintegerprogrammingtechniques,constraintprogrammingtechniques,andlo-calsearchtechniques.Itseemsthatinthecollegebasketballconferenceex-ampletheconstraintprogrammingtechniquehasbeenmoreeffectivethantheintegerprogrammingtechnique.However,itstillwouldbeinterestingtofindoutwhenitisappropriatetouseanoptimizationtechnique,whenaconstraintprogrammingtechnique,andwhenalocalsearchtechnique.Op-timizationtechniquesmaybemoresuitablewhenthereisaclearlydefinedobjectivefunctionandthesetoffeasiblesolutionsislarge.Anotherquestionmaybeofinterestaswell:aretherecasesinwhichitwouldmakesensetouseahybridapproachthatincludesoptimization,constraintprogramming,andlocalsearch?Whatisthebestwaytoincorporatethethreeapproacheswithinasingleframework?Moreresearchisneededinordertogetsomeinsightinhowthetechniqueshavetobecombinedinordertomaximizetheoveralleffectiveness.TheframeworkofAlgorithm10.2.2isbasedonanapproachthatfirstde-cidesontheHAPsandthenassignsteamstothedifferentHAPs.Acompletelydifferentapproachfirstassignsteamstogames,andthendeterminesthehome-awayassignmentswhileattemptingtominimizethenumberofbreaks(seeExercise10.7).
10.7Discussion251Sinceeachframeworkfordealingwithatournamentschedulingproblemconsistsofmultiplesteps,itmaybethecasethatinsomestepsitisap-propriatetouseacustomizedheuristic.Therehavebeenanumberofappli-cationsofcustomizedheuristics(e.g.,localsearch)intournamentschedul-ing.Inpractice,therearemanyside-constraintsintournamentscheduling.Ba-sically,thedatesonwhichsomeofthegamescanbeplayedarefixedinad-vance.Theymayalsobeinfluencedbyoutsideconsiderationssuchasnationalholidays,longweekends,andsoon.Also,thescheduleofoneteammayde-pendverystronglyonthescheduleofanotherteam.Forexample,inEuropeanumberofbigcitieshavemorethanonetopteamintheirnationalsoccerleague(e.g.,MilanhasACandInter,MadridhasRealandAtletico,andsoon)andtheseteamsmayhavefansincommon.Itisdesirablethatascheduledoesnothavethehomegamesoftwosuchteamsinthesameweekend.Ideally,whenoneclubplaysathome,theothershouldbeaway.Therearesimilaritiesaswellasdifferencesbetweentournamentschedulingandtheschedulingoftelevisionprograms.Intournaments,gameshavetobeassignedtoslotsandintelevisionprogramsshowshavetobeassignedtoslotsaswell.However,thereasonwhytournamentschedulingtendstobeaharderproblemisbasedonthefactthatagameinvolvestwoteamsandtheschedulesofbothteamshavetobetakenintoconsideration.Ontheotherhand,theassignmentofatelevisionshowtoaslotissomewhat(butnotcompletely)independentoftheassignmentofothershowstootherslots.Theremaybesomedependencybecauseoflead-ineffects.Thedependencybetweentheratingsoftwoconsecutiveshowscanbecomparedtohavingoneteamfacetwoverystrongteamsinarow.Besidestheschedulingoftheshowsthathavetobebroadcasted,therearealsoschedulingproblemsconcerningthecommercialsfromadvertisers.Largecompaniesthataremajoradvertisersbuyhundredsoftimeslotsfromanetworktoaircommercialsduringabroadcastseason.Theactualcommercialstobeassignedtotheseslotsaredeterminedatalaterstage.Duringthebroadcastseasontheclientsshipthevideotapesofthecommercialstobeairedintheslotstheypurchased.Theadvertisersoftenspecifythefollowingguideline:wheneveracommercialhastobeairedmultipletimeswithinagivenperiod,sayamonth,theadvertisementsshouldbespacedoutasevenlyaspossibleoverthatperiod.Thequestionthenarises:howtoassignthecommercialstotheslotsinsuchawaythatairingsofthesamecommercialarespacedasevenlyaspossible.Therearealsosomesimilaritiesbetweentournamentschedulingandtheschedulingofcommercials.Forexample,commercialsfromthesameadvertiserhavetobedistributedasevenlyaspossibleovertheplanninghorizonandgamesbetweenthesamepairsofteamshavetobespacedoutasevenlyaspossibleoverthedurationofthetournament.
25210SchedulingandTimetablinginSportsandEntertainmentExercises10.1.ExplainwhyitisnecessarytogeneratendifferentHAPsinStep1ofAlgorithm10.2.2.Thatis,whyisitnotpossiblefortwoteamstohavethesameHAP?10.2.GiveanexampleofapairofdifferentHAPs(eachonecontainingoneBye)thatcannotappeartogetherinthesamepatternset.Explainwhy.CanyougiveanexampleofapairofHAPs(withneitheronecontaininganyByes)thatcannotappearinthesamepatternset?Explainwhyorwhynot.10.3.ShowwhyinthegraphcoloringproblemdescribedinSection10.2thetwosubgraphsthatcorrespondtotwoconsecutiveroundshaveaHamiltonianpathwhenthereisnobreakinthesecondoneofthetworounds.Discussthestructureofthepathforthecasewhennisevenaswellasforthecasewhennisodd.10.4.ConsiderthetournamentschedulegeneratedinExample10.2.1.Thenumberofbreaksinthefinalscheduleis6.Canyoucreateaschedulewithlessbreaks?10.5.ConsiderthetournamentscheduleinExample10.2.3.Theminimumdistancebetweenanytwogamesbetweenthesamepairofteamsis3.Showthatitisimpossibletocreateascheduleinwhichtheminimumdistanceismorethan3.10.6.Describethesingleroundrobintournamentschedulingproblemasaworkforceconstrainedprojectschedulingproblem.10.7.Considerasituationinwhichsomeofthegameshavealreadybeenassignedtospecifictimeslots.However,ofthosegamesthatalreadyhavebeenassignedtogivenslots,ithasnotbeendeterminedyetwhichteamplaysathomeandwhichteamplaysaway.slots12345teama:bfcteamb:afteamc:deateamd:ceteame:fdcteamf:eab(a)Completethepartialschedulepresentedabovewithouttakingintoconsiderationwhethergamesareathomeoraway.Isthistypeofcompletionproblemaneasyproblemorahardproblem?(b)Withthescheduleofgamesdevelopedunder(a),assigntoeachgameahometeamandanawayteaminsuchawaythatthetotalnumberofbreaksisminimized.
CommentsandReferences25310.8.ConsidertheTravellingTournamentproblemdescribedinExample10.4.3.Thetraveltimematrixbetweenthesitesofthe6teamsissymmetricandgivenbelow.teams123456team1:-37205team2:3-4635team3:74-714team4:267-51team5:0315-2team6:55412-(a)ComputethevalueoftheobjectivefunctionoftheschedulepresentedinExample10.4.3assumingthatα=10.(b)Applyeachoneofthethreetypesofmovesdescribed(i.e.,SwapSlots,SwapTeams,andSwapHomes)tothesolutiongiveninExample10.4.3.De-scribehowthenewschedulesaregeneratedandcomputethenewvaluesoftheobjectivefunction.10.9.ConsiderthenetworktelevisionschedulingproblemdescribedinSection10.5.(a)Assumethateachshowtakesexactlyoneslot(halfanhour)andthattherearenolead-ineffects.Showthattheproblemthenreducestoanassign-mentproblem.(b)Considernowthecasewheresomeshowstaketwoslotsandthere-mainingshowstakeoneslot(again,nolead-ineffects).Istheproblemstillanassignmentproblem?CommentsandReferencesAsignificantamountofworkhasbeendoneontimetablingofsporttournaments.DeWerra(1988)studiedthegraphtheoreticpropertiesoftournamentscheduling.ThebreakminimizationproblemhasbeenstudiedbyMiyashiroandMatsui(2003).TheapproachdescribedinSection10.2whichfirstselectstheHAPsandthenassignsteamstoHAPshasbeenusedbyNemhauserandTrick(1998)andSchreuder(1992).Analternativeapproach,whichfirstassignsteamstogamesandthendeterminesthehome-awayassignment,hasbeenstudiedbyTrick(2001).Section10.3isbasedontheworkbyHenz(2001).Manyotherresearchershavealsoappliedconstraintprogrammingtechniquestotournamentscheduling;see,forexample,McAloon,TretkoffandWetzel(1997),Schaerf(1999),R´egin(1999),Henz,M¨uller,andThiel(2004),AggounandVazacopoulos(2004).ThefirstpartofSection10.4isbasedontheworkbyHamiezandHao(2001);theyappliedtabu-searchtotournamentscheduling.ThesecondpartofSection10.4,theTravelingTournamentProblem,isbasedontheworkbyAnagnostopoulos,Michel,VanHentenryck,andVergados(2003).Anumberofotherresearchershave
25410SchedulingandTimetablinginSportsandEntertainmentappliedlocalsearchtechniquestotournamentschedulingproblems;see,forexample,Sch¨onberger,Mattfeld,andKopfer(2004)whoappliedvarioustypesoflocalsearchalgorithmstothetimetablingofnon-commercialsportleagues.Section10.5,whichfocusedonthetimetablingoftelevisionshows,isbasedontheworkbyHoren(1980)andReddy,Aronson,andStam(1998).Forotherworkconcerningschedulinginbroadcastnetworks,seeHall,LiuandSidney(1998),Bol-lapragada,Cheng,Phillips,Garbiras,Scholes,Gibbs,andHumphreville(2002).Forworkconcerningtheschedulingofcommercialsinbroadcasttelevision,seeBollapra-gadaandGarbiras(2004),Bollapragada,Bussieck,andMallik(2004)andH¨agele,D´unlaing,andRiis(2001).TheschedulingofthecollegebasketballconferencedescribedinSection10.6isbasedonthepapersbyNemhauserandTrick(1998)andHenz(2001).AninterestingcaseconcerningtheschedulingofsoccerteamsintheNetherlandswasanalyzedbySchreuder(1992).Bartsch,DrexlandKr¨oger(2004)developedschedulesfortheprofessionalsoccerleaguesofAustriaandGermany.
Chapter11Planning,Scheduling,andTimetablinginTransportation11.1Introduction……………………………25511.2TankerScheduling……………………….25611.3AircraftRoutingandScheduling……………26011.4TrainTimetabling……………………….27411.5JeppesenSystems:DesignandImplementation.28111.6Discussion……………………………..28511.1IntroductionInthetransportationindustryplanningandschedulingproblemsabound.Thevarietyintheproblemsisduetothemanymodesoftransportation,e.g.,shipping,airlines,andrailroads.Eachmodeoftransportationhasitsownsetofcharacteristics.Theequipmentandresourcesinvolved,i.e.,(i)shipsandports,(ii)planesandairports,and(iii)trains,tracks,andrailwaystations,havedifferentcostcharacteristics,differentlevelsofflexibilities,anddifferentplanninghorizons.Thesecondsectionofthischapterfocusesonoiltankerscheduling.Thesemodelsareusedinpracticeinarollinghorizonmanner.Amongallmodelsdiscussedinthischapter,thisoneistheeasiesttoformulate.Thesubse-quentsectionconsidersaircraftroutingandscheduling.Inaircraftroutingandschedulingthegoalistocreateaperiodic(daily)timetable.Inacertainsensethismodelisanextensionofthemodelforoiltankerscheduling.Thein-tegerprogrammingformulationoftheaircraftroutingandschedulingproblemisverysimilartotheformulationdescribedfortheoiltankerschedulingprob-lem;however,intheairlinecasethereareadditionalconstraintsthatenforce© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_11,255
25611Planning,Scheduling,andTimetablinginTransportationperiodicity.Thefourthsectiondiscussestimetablingoftrains.Trackcapacityconstraintsinrailwayoperationsspecifythatonetraincanpassanotheronlyatastation,andnotinbetweenstations.Thefifthsectiondescribestheair-lineroutingandschedulingsystemsdesignedandimplementedbyJeppesenSystems.Thediscussionsectionfocusesonthesimilaritiesanddifferencesbe-tweentankerscheduling,airlineroutingandscheduling,andtraintimetabling.11.2TankerSchedulingCompaniesthatownandoperatetankerfleetstypicallymakeadistinctionbetweentwotypesofships.Onetypeofshipiscompany-ownedandtheothertypeofshipischartered.Theoperatingcostofacompany-ownedshipisdifferentfromthecostofacharterthatistypicallydeterminedonthespotmarket.Eachshiphasaspecificcapacity,agivendraught,arangeofpossiblespeedsandfuelconsumptions,andagivenlocationandtimeatwhichtheshipisreadytostartanewtrip.Eachportalsohasitsowncharacteristics.Portrestrictionstaketheformoflimitsonthedeadweight,draught,length,beamandotherphysicalchar-acteristicsoftheships.Theremaybesomeadditionalgovernmentrulesineffect;forexample,theNigeriangovernmentimposesaso-called90%rulewhichstatesthatalltankersmustbeloadedtomorethan90%ofcapacitybeforesailing.Acargothathastobetransportedischaracterizedbyitstype(e.g.,typeofcrude),quantity,loadport,deliveryport,timewindowconstraintsontheloadanddeliverytimes,andtheloadandunloadtimes.Ascheduleforashipdefinesacompleteitinerary,listinginsequencetheportsvisitedwithinthetimehorizon,thetimeofentryateachportandthecargoesloadedordeliveredateachport.Theobjectivetypicallyistominimizethetotalcostoftransportingallcargoes.Thistotalcostconsistsofanumberofelements,namelytheoperatingcostsforthecompany-ownedships,thespotcharterrates,thefuelcosts,andtheportcharges.Portchargesvarygreatlybetweenportsandwithinagivenportchargestypicallyvaryproportionallywiththedeadweightoftheship.Inordertopresentaformaldescriptionoftheproblemthefollowingno-tationisused.Letndenotethenumberofcargoestobetransported,Tthenumberofcompany-ownedtankers,andpthenumberofports.LetSidenotethesetofallpossibleschedulesforshipi.Schedulelforshipi,l∈Si,isrepresentedbythecolumnvectorali1ali2…alin
11.2TankerScheduling257Theconstantalijis1ifunderschedulelshipitransportscargojand0other-wise.Letclidenotetheincrementalcostofoperatingacompany-ownedshipiunderschedulelversuskeepingshipiidleovertheentireplanninghorizon.Theoperatingcostcanbecomputedonceschedulelhasbeenspecified,sinceitmaydependinvariouswaysonthecharacteristicsoftheshipandoftheschedule,includingthedistancetravelled,thetimetheshipisused,andtheportsvisited.Thecostc∗jdenotestheamountthathastobepaidonthespotmarkettotransportcargojonashipthatisnotcompanyowned.Letπli=nj=1alijc∗j−clidenotethe”profit”(i.e.,theamountofmoneythatdoesnothavetobepaidonthespotmarket)byoperatingshipiaccordingtoschedulel.Thedecisionvariablexliis1ifshipifollowsschedulelandzerootherwise.TheTankerSchedulingProblemcannowbeformulatedasfollows:maximizeTi=1l∈SiπlixlisubjecttoTi=1l∈Sialijxli≤1j=1,…,nl∈Sixli≤1i=1,…,Txli∈{0,1}l∈Si,i=1,…,TTheobjectivefunctionspecifiesthatthetotalprofithastobemaximized.Thefirstsetofconstraintsimplythateachcargocanbeassignedtoatmostonetanker.Thesecondsetofconstraintsspecifiesthateachtankercanbeassignedatmostoneschedule.Theremainingconstraintsimplythatalldeci-sionvariableshavetobebinary0−1.Thisoptimizationproblemistypicallyreferredtoasaset-packingproblem.Thealgorithmusedtosolvethisproblemisabranch-and-boundproce-dure.However,beforethebranch-and-boundprocedureisapplied,acollec-tionofcandidatescheduleshavetobegeneratedforeachshipinthefleet.Asstatedbefore,suchaschedulespecifiesanitineraryforaship,listingtheportsvisitedandthecargoesloadedordeliveredateachport.Thegen-erationofaninitialcollectionofcandidatescheduleshastobedonebyaseparatead-hocheuristicthatisespeciallydesignedforthispurpose.Thecollectionofcandidateschedulesshouldincludeenoughschedulessothatpo-tentiallyoptimalschedulesarenotignored,butnotsomanythattheset-
25811Planning,Scheduling,andTimetablinginTransportationpackingproblembecomesintractable.Physicalconstraintssuchasshipca-pacityandspeed,portdepthandtimewindowslimitthenumberoffeasiblecandidateschedulesconsiderably.Schedulesthathaveanegativeprofitcoeffi-cientintheobjectivefunctionoftheset-packingformulationcanbeomittedaswell.Thebranch-and-boundmethodforsolvingtheproblemistypicallybasedoncustomizedbranchingandboundingprocedures.Sincetheproblemisamaximizationproblemagoodschedulegeneratedbyacleverheuristic(oramanualmethod)providesalowerboundforthevalueoftheoptimalsolution.Whenconsideringaparticularnodeinthebranchingtree,itisnecessarytodevelopanupperboundforthecollectionofschedulesthatcorrespondtoallthedescendantsofthisparticularnode;ifthisupperboundislessthanthelowerboundontheoptimumprovidedbythebestschedulecurrentlyavailable,thenthisnodecanbefathomed.Thereareavarietyofsuitablebranchingmechanismsforthebranch-and-boundtree.Thesimplestmechanismisjustthemostbasic0−1branch-ing.Selectatanodeavariablexliwhichhasnotbeenfixedyetatahigherlevelnodeandgeneratebranchestotwonodesatthenextleveldown:onebranchforxli=0andoneforxli=1.TheselectionofthevariablexlimaydependonthesolutionoftheLPrelaxationatthatnode;themostsuitablexlimaybetheonewithavalueclosestto0.5inthesolutionoftheLPrelaxation.Ifatanodeavariablexliissetequalto1forshipi,thencertainschedulesforothershipscanberuledoutforallthedescen-dantsofthisnode;thatis,theschedulesforothershipsthathaveacargoincommonwithschedulelforshipidonothavetobeconsideredanymore.Anotherwayofbranchingcanbedoneasfollows:SelectatagivennodeashipithathasnotbeenselectedyetatahigherlevelnodeandgenerateforeachschedulelinSiabranchtoanodeatthenextleveldown.Inthebranchcorrespondingtoschedulelthevariablexli=1.Usingthisbranchingmechanism,onestillhastodecideateachnodewhichshipitoselect.Onecouldselecttheibasedonseveralcriteria.Forexample,ashipthattransportsmanycargoesorashipthatmayberesponsibleforalargeprofit.AnotherwayistoselectanithathasahighlyfractionalsolutionintheLPrelaxationoftheproblem(e.g.,theremaybeashipiwithasolutionxli=1/KforKdifferentscheduleswithKbeingafairlylargenumber).Anupperboundatanodecanbeobtainedbysolvingthelinearrelaxationoftheset-packingproblemcorrespondingtothatnode,i.e.,theintegralityconstraintsonxliarereplacedbythenonnegativityconstraintsxli≥0.Thisproblemmaybereferredtoasthecontinuousset-packingproblem.Thevalueoftheoptimalsolutionisanupperboundforthevaluesofallpossibleso-lutionsoftheset-packingproblematthatnode.Itisnowadayspossibletofindwithlittlecomputationaleffortoptimalsolutions(oratleastgoodup-perbounds)forverylargecontinuousset-packingproblems,makingsuchaboundingmechanismquiteeffective.
11.2TankerScheduling259Example11.2.1(OilTankerScheduling).Considerthreeshipsand12cargoesthathavetobetransported.Afeasibilityanalysisshowsthatforeachoneoftheshipstherearefivefeasibleschedules.The15columnsinthetablebelowrepresentthe15feasibleschedulesforthethreeships.Schedulesa11ja21ja31ja41ja51ja12ja22ja32ja42ja52ja13ja23ja33ja43ja53jcargo1100110100000010cargo2100001000001011cargo3001010001100000cargo4011101010000000cargo5110000001000101cargo6000110100110000cargo7000000011000001cargo8010001011100000cargo9001000100111100cargo10010001000011000cargo11000000110001110cargo12000100000010111Ifacargoistransportedbyacharter,thenachartercostisincurred.Cargoes123456789101112CharterCosts1429132312085122173221717751885246819281634741Theoperatingcostsofthetankersundereachoneoftheschedulesaretabulatedbelow:Schedulel12345costoftanker1(cl1)56585033272235053996costoftanker2(cl2)40196914469379106868costoftanker3(cl3)58295588828433384715Theprofitsofeachschedulecannowbecomputed.Schedulel12345profitoftanker1(πl1)−733146514661394858profitoftanker2(πl2)16298341113−869910profitoftanker3(πl3)15251765−126817891297Theintegerprogramcannowbeformulatedasfollows:maximize−773×11+1465×21+1466×31+1394×41+858×51+1629×12+834×22+1113×32−869×42+910×52+1525×13+1765×23−1268×33+1789×43+1297×53
26011Planning,Scheduling,andTimetablinginTransportationsubjecttox11+x41+x51+x22+x43≤1×11+x12+x23+x43+x53≤1×31+x51+x42+x52≤1×21+x31+x41+x12+x32≤1×11+x21+x42+x33+x53≤1×41+x51+x22+x52+x13≤1×32+x42+x53≤1×21+x12+x32+x42+x52≤1×31+x22+x52+x13+x23+x33≤1×21+x12+x13+x23≤1×22+x32+x23+x33+x43≤1×41+x13+x33+x43+x53≤1×11+x21+x31+x41+x51≤1×12+x22+x32+x42+x52≤1×13+x23+x33+x43+x53≤1xli∈{0,1}Aninitialupperboundcanbeobtainedbysolvingthelinearrelaxationoftheintegerprogram,i.e.,allowingxlitoassumeanyvaluebetween0and1.Thesolutionofthelinearprogramisx21=x31=x51=1/3,×12=x52=1/3,andx13=1/3,×43=2/3.Thevalueofthesolution(i.e.,theupperbound)is3810.33.Solvingthisintegerprogramviabranch-and-boundresultsinthetreeshowninFigure11.1.Theoptimalsolutionoftheintegerprogramassignsschedule3toship1(i.e.,x31=1),andschedule4toship3(i.e.,x43=1).Ship2remainsidleandcargoes5,6,7,8,and10aretransportedbycharters.Thevalueofthissolutionis3255.Incontrasttothetransportationproblemsanalyzedinthenexttwosec-tions,schedulesfortankers(oil,naturalgas,bulkcargoingeneral)areusuallynotcyclic.Theschedulingprocessisbasedonarollinghorizonprocedure.11.3AircraftRoutingandSchedulingAmajorproblemfacedbyeveryairlineistoconstructadailyscheduleforaheterogeneousaircraftfleet.Aplanescheduleconsistsofasequenceofflightlegsthathavetobeflownbyaplanewiththeexacttimesatwhichthelegs
11.3AircraftRoutingandScheduling261Subproblem1 ( t = 1 )235111152214331/31/31/3,2/3UB3810.33LB2290xxxxxxx========>=Subproblem2 ( t = 2 )351113522214331/31/31/3,2/3UB3693LB2290xxxxxxx========>=Subproblem3 ( t = 13 )24131,1UB3254LB3255xx===<=210x=211x=Subproblem4 ( t = 3 )51312214330.5,0.50.5UB3457LB2290xxxxx======>=Subproblem6 ( t = 4 )413512221243331/3, 1/3UB3375LB2290xxxxxxx========>=310x=510x=Subproblem7 ( t = 9 )512512331,1/3,1/3UB3163.667 LB3159xxxx=====>=Subproblem5 ( t = 12 )34131, 1UB3255 > LB3159xx====Candidate solution (optimal)Update LB=3255311x=511x=UB < LBAlgorithm terminates here Subproblem8 ( t = 5 )132214331/21/2UB3028LB2290xxxx=====>=Subproblem9 ( t = 8 )42131, 1UB3159LB2699xx===>=410x=Subproblem10 ( t = 6 )54231, 1UB2699LB2290xx===>=120x=Subproblem11 ( t = 7 )121UB1629LB2699x==<=121x=411x=Subproblem12 ( t = 10 )513122531, 1/2UB2877.5LB3159xxxx=====<=Subproblem13 ( t = 11 )52131, 1UB2623LB3159xx===<=230x=231x=Candidate solutionLB not updated Candidate solutionUpdate LB=3159 Candidate solutionUpdate LB=2699Candidate SolutionLB not updated UB < LBThis node is deleted1/31/2Fig.11.1.Branch-and-boundtreefortankerschedulingproblem 26211Planning,Scheduling,andTimetablinginTransportationmuststartandfinishattherespectiveairports.Thefirstpartoftheprob-lem(determiningthesequenceofflightlegs)isbasicallyaroutingproblem,whereasthesecondpartoftheproblem(determiningtheexacttimes)isaschedulingproblem.Thefleetscheduleisimportant,sincethetotalrevenueoftheairlinecanbeestimatedifthedemandfunctionofeachlegisknown.Moreover,thefleetschedulealsodeterminesthetotalcostincurredbytheairline,includingthecostoffuelandthesalariesofthecrews.Anairlinetypicallyhas,frompastexperienceandthroughmarketingre-search,estimatesofcustomerdemandsforspecificflightlegs(aflightlegischaracterizedbyitspointoforigin,itsdeparturetimeanditsdestination).Itcanbeassumedthataminorshiftinthedeparturetimeoftheflightlegdoesnothaveaneffectonthedemand.Soanairlinehasforeachlega(narrow)timewindowinwhichitcandepart.Anairlinealsohasestimatesfortherevenuederivedfromaspecificflightlegasafunctionofthetypeofplaneutilized,andofthecostsinvolved.TheDailyAircraftRoutingandSchedulingProblemcannowbeformu-latedasfollows:Givenaheterogeneousaircraftfleet,acollectionofflightlegsthathavetobeflowninaone-dayperiodwithdeparturetimewindows,du-rations,andcost/revenuescorrespondingtotheaircrafttypeforeachleg,afleetschedulehastobegeneratedthatmaximizestheairline’sprofits(possiblysubjecttocertainadditionalconstraints).SomeoftheadditionalconstraintsthatoftenhavetobetakenintoaccountinanAircraftRoutingandSchedulingProblemarethenumberofavailableplanesofeachtype,therestrictionsoncertainaircrafttypesatcertaintimesandatcertainairports,therequiredconnectionsbetweenflightlegs(theso-called”thrus”)imposedbytheairlineandthelimitsonthedailyserviceatcertainairports.Also,thecollectionofflightlegsmayhavetobebalanced,i.e.,ateachairporttheremustbe,foreachairplanetype,asmanyarrivalsasdepartures.Onemustfurtherimposeateachairporttheavailabilityofanequalnumberofaircraftofeachtypeatthebeginningandattheendoftheday.Intheformulationoftheproblemthefollowingnotationisused:Ldenotesthesetofflightlegs,Tdenotesthenumberofdifferentaircrafttypes,andmidenotesthenumberofavailableaircraftoftypei,i=1,...,T.SothetotalnumberofaircraftavailableisTi=1mi.(NotethatthisisincontrasttotheprevioussectionwhereTwasthetotalnumberoftankers.)Someflightlegsmaybeflownbymorethanonetypeofaircraft.LetLidenotethesetofflightlegsthatcanbeflownbyanaircraftoftypeiandletSidenotethesetoffeasibleschedulesforanaircraftoftypei.Thissetincludestheemptyschedule(0);anaircraftassignedtothisscheduleissimplynotbeingused.Letπijdenotetheprofitgeneratedbycoveringflightlegjwithanaircraftoftypei.Witheachschedulel∈Sithereisa 11.3AircraftRoutingandScheduling263totalanticipatedprofitπli=j∈Liπijalij,wherealijis1ifschedulelcoverslegjand0otherwise.Ifanaircrafthasbeenassignedtoanemptyschedule,thentheprofitisπ0i.Theprofitπ0imaybeeithernegativeorpositive.Itmaybenegativewhenthereisahighfixedcostwithkeepingaplaneforaday;itmaybepositivewhenthereisabenefitwithhavingaplaneidle(someairlineswanttohaveidleplanesthatcanserveasstandby).LetAdenotethesetofairports,andAibethesubsetofairportsthathavefacilitiestoaccomodateaircraftoftypei.Letolihbeequalto1iftheoriginofschedulel,l∈Si,isairporth,and0otherwise;letdlihbeequalto1ifthefinaldestinationofschedulelisairporth,and0otherwise.Thebinarydecisionvariablexlitakesthevalue1ifschedulelisassignedtoanaircraftoftypei,and0otherwise;theintegerdecisionvariablex0idenotesthenumberofunusedaircraftoftypei,i.e.,theaircraftthathavebeenassignedtoanemptyschedule.TheDailyAircraftRoutingandSchedulingProblemcannowbeformu-latedasfollows:maximizeTi=1l∈SiπlixlisubjecttoTi=1l∈Sialijxli=1j∈Ll∈Sixli=mii=1,...,Tl∈Si(dlih−olih)xli=0i=1,...,T,h∈Aixli∈{0,1}i=1,...,T,l∈SiTheobjectivefunctionspecifiesthatthetotalanticipatedprofithastobemaximized.Thefirstsetofconstraintsimplythateachflightleghastobecoveredexactlyonce.(Thissetofconstraintsissomewhatsimilartothefirstsetofconstraintsintheformulationofthetankerschedulingproblem.)Thesecondsetofconstraintsspecifiesthemaximumnumberofaircraftofeachtypethatcanbeused.Thethirdsetofconstraintscorrespondtotheflowconservationconstraintsatthebeginningandattheendofthedayateachairportforeachaircrafttype.Theremainingconstraintsimplythatalldecisionvariableshavetobebinary0−1. 26411Planning,Scheduling,andTimetablinginTransportationThismodelisbasicallyaSetPartitioningProblemwithadditionalcon-straints;thetankerschedulingmodeloftheprevioussectionisaSetPackingproblembecauseofthefactthatthefirstsetofconstraintsareinequality(≤)constraints(seeAppendixA).Thealgorithmtosolvethisproblemisalsobasedonbranch-and-bound;theversionofbranch-and-boundistypicallyre-ferredtoasbranch-and-price(seeAppendixB).Wefirstdescribethemecha-nismforgeneratingupperboundsandthendescribethebranchingstrategies.Upperboundscanbeobtainedbyusingaso-calledcolumngenerationprocedurethatsolvesthelinearrelaxationoftheintegerprogramformulatedabove(i.e.,inthelinearrelaxationtheintegralityconstraintsonthedecisionvariablesxliarereplacedbynonnegativityconstraintsxli≥0;becausethefirstsetofconstraintsalreadyensurethatthedecisionvariablesxlihavetobelessthanorequalto1).Thecolumngenerationprocedureisusedtoavoidthenecessityofgener-atingallpossibleschedules.Theproceduredividesthelinearprogramintoarestrictedmasterproblemandasubproblem.Therestrictedmasterproblemisbasicallyalinearprogramdefinedoverarelativelysmallnumberofcandi-dateschedulesfortheaircraft.Thedecisionvariablexlithatcorrespondstoacolumnthathasnotbeenincludedisassumedtobe0.Theideaistofindanoptimalsolutionforthecurrentrestrictedmasterproblemandcomputethedualvariablesassociatedwiththissolution.Thesedualvariablesrepresenttheimputedunitcostsforthe”resources”,suchasthelegs,theaircraftandtheairports.Thesevariablesareusedinthesubproblemtocomputethepoten-tialprofitofothercandidateschedules.Thesubproblemisbasicallyusedtotestwhetherthesolutionofthecurrentrestrictedmasterproblemisoptimaloverallpossibleschedules,i.e.,fortheunrestricted(butstilllinear)masterproblem.Thesubproblem,whichusesthedualvariablesoftheoptimalso-lutionforthecurrentrestrictedmasterproblem,turnsouttobeequivalenttoalongestpathproblemwithtimewindows;itcanbesolvedbydynamicprogramming.Ifthecurrentsolutionisnotoptimaloverallschedules(i.e.,therearescheduleswithapositivepotentialprofitinthesubproblem),thenthesubproblemmustprovideoneormorenewaircraftschedulestobeaddedtothesetofcandidateschedulesintherestrictedmasterproblem.Thesubproblemforplanesoftypeicanbeformulatedasfollows.ConsideradirectedgraphGi=(Ni,Bi)thatisusedtogeneratenewfeasiblecandidateschedulesforanaircraftoftypei,seeFigure11.2.Therearefivetypesofnodes:onesource,onesink,originationairportnodes,terminationairportnodes,andflightlegnodes.TherearefivetypesofarcsinBi:sourcearcs,sinkarcs,scheduleoriginationarcs,scheduleterminationarcs,andturnarcs.Asourcearcgoesfromthesourcetoanoriginationairportnode.Asinkarcgoesfromaterminationairportnodetothesink.Ascheduleoriginationarcemanatesfromanorig-inationairportnodeandgoestoaflightlegnode(theflighthastostartat 11.3AircraftRoutingandScheduling265SourceOriginationAirportNodesFlight Nodes(O)TerminationAirportNodesSinkSinkArcsSinkArcsScheduleOriginationArcsScheduleTerminationArcsTurn ArcsFig.11.2.Graphforgeneratingfeasibleaircraftschedules 26611Planning,Scheduling,andTimetablinginTransportationtheairportassociatedwiththeoriginationairportnode).Ascheduletermi-nationarcbeginsataflightlegnodeandgoestoaterminationairportnode(theflighthastoendattheairportassociatedwiththeterminationairportnode).Aturnarcconnectstwoflightlegnodesandsimplyrepresentsacon-nectionbetweenthesetwoflightlegs.Suchanarcexistsbetweentwoflightlegnodesonlyiftheflightlegscanbeflownbythesameaircraftconsecutivelywhilerespectingthetimewindowsofeachflightlegandthenecessarytimeittakestoturntheaircraftaroundinbetweentheflightlegs.(Aminimumturnaroundtimeisneededinordertodeplaneonegroupofpassengers,cleantheplaneandboardthedepartingpassengers).Sotheexistenceofaturnarcbetweentwoflightnodesdependsonthegiventimewindowsforthede-parturetimesofthetwoflightlegsaswellasontheminimumturnaroundtime.Theobjectiveofthesubproblemistofindafeasibleplaneschedulewithmaximummarginalprofit.Tobefeasible,aschedulemustsatisfythetimewindowconstraintsconcerningdeparturetimes.Ifeijistheearliestpossibledeparturetimeofflightlegjandijisthelatestpossibledeparturetime,thenthevalueoftheactualdeparturetimevariablemustliewithinthetimewindow[eij,ij].Ifτijisthedurationoflegjandδijkistheminimumturnaroundtimebetweenlegsjandk,thenforaturnarctoexistbetweenlegsjandk,eij+τij+δijk≤ikEachpathfromthesourcenodetothesinknodecorrespondstoafeasibleschedule.Themarginalprofitofaschedulecannowbecomputedusingtherevenueandthecostofeachactivity(i.e.,theprofit)aswellasthedualvariablesassociatedwiththecurrentoptimalsolutionoftherestrictedmasterproblem.ThearcsingraphGiareassignedvaluescorrespondingtotheprofitsoftheactivitiesandthedualvariablesassociatedwiththeoptimalsolutionofthecurrentrestrictedmasterproblem.Everyarcleavingflightnodejisassignedaprofitπij(whichisaknownquantity).Inadditiontotheprofits,thedualvariableshavetobeplacedonspecificarcsinthenetwork.Letαj,βiandγihdenotethedualvariablesassociatedwiththefirst,secondandthirdsetofconstraintsintherestrictedmasterproblem.Adualvariablerepresentstheincreaseinthetotalprofitofthecurrentsolutionoftherestrictedmasterproblemassumingthattherighthandsideofthecorrespondingconstraintisincreasedbyoneunit.Soifthenumberofaircraftoftypeiisincreasedby1,thentheprofitgoesupbyβi.Ifthereareanumberofunusedaircraftinthesolutionofthecurrentmasterproblem,thenβi=0.Anotherinterpretationofthedualvariableβiisthefollowing:ifinthecurrentsolutionoftherestrictedmasterproblemallaircraftarebeingused,thenassigningascheduleltoanaircraftimpliesacostβi(sincethatisthecurrentopportunitycostofaplane).Thepotentialprofit¯πliofschedulel∈Siwithrespecttothecurrentsolutioncannowbecomputedasitistheprofitofthecorrespondingpathinthenetwork(schedulelmayormaynotbepartoftherestrictedmaster 11.3AircraftRoutingandScheduling267problem).Thepotentialprofitis¯πli=j∈Li(πij−αj)alij−βi−h∈Aiγih(dlih−olih),whereπijistheprofitgeneratedbyflyinglegjwithanaircraftoftypei.Theαjisthecurrentcostincurredbyoperatinglegj,theβiisthecurrentcostincurredbyusinganadditionalaircraftoftypei,andtheγihisthecurrentcostincurredbyallowinganimbalanceofoneairplaneoftypeiatairporth.IngraphGi,thepotentialprofitisequaltothesumoftheprofitsonthearcsformingthepathcorrespondingtoschedulel(negativeprofitsbeingcosts).Summarizing,inordertocomputethepotentialprofitofaschedulel∈Si,thefollowingvaluesmustbeassignedtothearcsofGi:arctypeprofitsourcearcs−βisinkarcs0originationairportarcsleavingoriginationairporth+γihterminationairportarcsleavingflightnodejπij−αj−γihturnarcsemanatingfromflightnodejπij−αjByregardingtheprofitsas”distances”,alongestpathalgorithmcannowbeusedtofindthepathwiththelargestpotentialprofitinnetworkGi.Forthisparticularversionofthelongestpathproblemitiseasytodevelopanefficientalgorithm(sincethetimewindowsinthenetworkmakesurethatthepathscannotcycle).Ifthelargestpotentialprofitispositive,thenthisflightscheduleisincludedintherestrictedmasterproblem,whichisthensolvedagain.Iffornoneoftheaircrafttypesthereisapathwithapositivepotentialprofit,thenthecurrentsolutionisoptimaloverallschedulesandthegenerationofschedulesterminates.Afterthisdescriptionofthemechanismtogenerateupperboundswecon-siderthebranchingprocedure.Therearevarioustypesofbranchingstrategies.Thesimplestmechanismisagainthebasic0−1branching.Selectatanodeavariablexliwhichhasnotbeenfixedyetatahigherlevelandgeneratebranchestotwonodesatthenextleveldown:onebranchforxli=1andtheotherbranchforxli=0.Settingxli=1impliesthatinthecorrespond-ingnodealltheflightlegscoveredbythatschedulecanbedeletedinGi,foralli=1,...,T.Moreover,theconstraintsinthemasterproblemthatcoverthecorrespondingflightscanbedeletedalso;otherconstraintsmaybeadjustedaccordingly.Settingxli=0impliesthatatthatnodethecorrespond-ingcolumnmustbeignored.Therearewaystoensurethatinthesubproblemanotherpathisselected.Anotherissuethathastoberesolvedinthebranch-and-boundprocedureistheselectionofcolumnsthatshouldbeconsideredascandidateschedulesatanyparticularnode.Onemaywanttokeepallcolumnsthathavebeengeneratedsofaratallnodespreviouslyanalyzed.Or,onemaywanttoselect 26811Planning,Scheduling,andTimetablinginTransportationthroughsomeheuristicacollectionofpromisingscheduleswithapositivepotentialprofit.Example11.3.1(AircraftRoutingandScheduling).Considertwotypesofplanes,i.e.,T=2.Type1planesarewidebodiesandm1=2;type2planesarenarrowbodiesandm2=2.Twelvelegshavetobeflownbetweenfourair-ports.Thefourairportsare:h=1:SanFrancisco(SFO)h=2:LosAngeles(LAX)h=3:NewYork(NYC)h=4:Seattle(SEA)Foreachflightthepairofcitiesisgivenaswellasatimewindowinwhichtheplanehastoflyandthetype(s)ofplanethatcanbeused.Inthisexampleeachlegcanbeflownbyeithertypeofplaneandthetimeittakesisindependentofthetype.However,theprofitmadeonaflightdependsonthetypeassigned.Legj123456789101112cities1→21→22→12→11→41→44→14→13→13→11→31→3τ1j1.51.51.51.533336666timea.m.p.m.a.m.p.m.a.m.p.m.a.m.p.m.a.m.p.m.a.m.p.m.Ana.m.flightmusttakeoffafter5a.m.andlandbefore1p.m.;ap.m.flightmusttakeoffafter1p.m.andlandbefore5a.m.thenextday.Basedontheseflightdataitcanbecheckedeasilywhetheraspecifictripforaplaneisfeasible.Initialsetsofcandidateschedulescanbegeneratedforbothtypesofplanes.AninitialsubsetS1ispresentedbelow.Schedulesa11ja21ja31ja41ja51ja61ja71ja81ja91ja101jflight10011000000flight20000000101flight30001000000flight40010000100flight50111000000flight60000100101flight70110101000flight80001000001flight90000010101flight100100111010flight110000111010flight120100011010AninitialsubsetS2isgiveninTable11.1.NotethatthefirstcandidatescheduleinbothS1andS2istheemptyscheduleinwhichtheplaneiskeptidle.TheolihanddlihvaluesofeachoneoftheseschedulescanbefoundinSection5oftheaccompanyingCD-ROM. 11.3AircraftRoutingandScheduling269Schedulesa12ja22ja32ja42ja52ja62ja72ja82ja92jflight1010100000flight2000000101flight3000100010flight4010000100flight5010000000flight6001100101flight7011001000flight8000100001flight9000010101flight10001011010flight11001011010flight12000011010Table11.1.TableforExample11.3.1.Theprofitsgeneratedbyassigninganaircraftoftypeitoflightlegjisπij.Theπijaretabulatedbelow.Legj123456789101112π1j4503005004009009009009001500150015001500π2j45045050050010001000100010001350135013501350Theprofitassociatedwiththeassignmentofschedule(orroundtrip)ltoaplaneoftypeiisπli.Forthelargeplanesoftype1theprofitsaretabulatedbelow.schedulel12345678910πl1−375480026502750480060005400310045003600Notethatkeepinganaircraftoftype1idleactuallycostsmoney.Forthesmallplanesoftype2theprofitsaretabulatedbelow.schedulel123456789πl2029504700295054005050330045503800Solvingthelinearrelaxationoftherestrictedmasterproblemresultsinanoptimalsolutionthatsetsx41=x71=x81=2/3andx12=1andx22=x82=x92=1/3.Theoptimalvalueoftheobjectivefunctionis11,267.Byapplyingastandardbranch-and-boundproceduretotheintegerpro-gram(seeFigure11.3),itcanbeverifiedthattheoptimalintegersolutionforthisrestrictedmasterproblemhasatotalprofitof10,725andthattheoptimalsolutionhasx11=x41=1andx62=x72=1. 27011Planning,Scheduling,andTimetablinginTransportation478111128922222311,311267xxxxxxxUB========47811116722211,211,211175xxxxxxUB=======131189221110625xxxxUB=====47111Infeasiblexx==146711221,110725Integer optimal solutionxxxxUB=====410x=411x=710x=711x=Fig.11.3.Branch-and-boundtreeforaircraftschedulingproblemInordertodeterminewhichschedules(roundtrips)arefeasibleandhaveapotentialprofitiftheywereincludedintherestrictedmasterproblem,thesubproblemhastobeformulated.Thedualvariables(orshadowprices)gen-eratedinthelinearprogrammingsolutionoftherestrictedmasterproblemarethefollowing:Legj123456789101112αj0−616.7216.7265025502733.3−2250015004333.30183.3Thedualvariablescorrespondingtothesecondsetofconstraintsareβ1=−16.67andβ2=0.(Itwastobeexpectedthattheβ2wouldbezerosinceoneofthesmallplanesiskeptidleintheoptimalsolution.)Thedualvariablescorrespondingtothethirdsetofconstraintsaretabulatedbelow: 11.3AircraftRoutingandScheduling271airporth1234γ1h0003150γ2h00183.332966.67Notethatthedualvariablesassociatedwiththeoptimalsolutionofthelinearprogrammingsolutionarenotnecessarilyauniqueset.Thedualvariablesgeneratedmaydependverymuchonthetypeoflinearprogrammingimple-mentation,seeExercise11.4.InthenetworkG1forwidebodiesanumberofcandidateschedulescanbegenerated.Thesecandidateschedulesinvolveflightlegs1,2,3,4,5,7,12.Thedurationsoftheseflightlegsandthetimewindowsduringwhichtheyhavetotakeplacearegiven.Legj12345712τ1j(hours)1.51.51.51.5336e1j5a.m.1p.m.5a.m.1p.m.5a.m.5a.m.1p.m.1j1p.m.5a.m.1p.m.5a.m.1p.m.1p.m.5a.m.Thecandidateschedulesinthetablebelowforthelargeplanesarefeasi-bleandofeachoneofthesecandidateschedulesthepotentialprofitcanbecomputed.Schedulesa111ja121ja131ja141ja151jflight101011flight200000flight311011flight400000flight500000flight601111flight700111flight801000flight900000flight1010101flight1110100flight1210001profitπl150002750480027505750dualvariables4716.672933.344800683.345200potentialprofit¯πl1283.33−183.3302066.67550GraphG1withschedules11and12isshowninFigure11.4.Aplaneoftype1thatfollowsschedule11startsoutinLosAngelesandendsupinNewYorkaftercoveringlegs3,11,10,12(inthatorder).Thatis,theplanefliesfromLosAngelestoSanFrancisco,thentoNewYork,backtoSanFranciscoandagaintoNewYork.Thepotentialprofitofschedule11is 27211Planning,Scheduling,andTimetablinginTransportationleg8leg6leg1SinkSourceleg11leg12leg10leg3Fig.11.4.GraphG1forplanesoftype1 11.3AircraftRoutingandScheduling273¯π111=(500−216.67)+(1500−0)+(1500−4333.33)+(1500−183.33)−16.67−0=283.33Aplanethatfollowsschedule12startsoutinSanFranciscoandendsupbackinSanFranciscoaftercoveringlegs1,3,6,8(inthatorder).However,thepotentialprofitofthisscheduleisnegative:¯π121=(450−0)+(500−216.67)+(900−2733.33)+(900−0)−16.67−0=−183.33Fromthetableitfollowsthatschedules11,14,and15shouldbeincludedinS1fortherestrictedmasterproblem.SointhenewrestrictedmasterproblemS1has10+3=13schedules.ForthenarrowbodiesanetworkG2hastobesetupandcandidatesched-uleshavetobeexamined.Thereare6additionalschedulesfornarrowbodiesthathavetoinvestigated.Theresultsaretabulatedbelow.Schedulesa102ja112ja122ja132ja142ja152jflight1010111flight2110001flight3010111flight4110001flight5110000flight6001110flight7101110flight8010000flight9000000flight10001010flight11001000flight12000010profitπl2295039004700295056501900dualvariables2233.3347004816.677005216.672250potentialprofit¯πl2716.67−800−116.672250433.33−350Itfollowsthatschedules10,13,and14havetobeincludedinS2.InthenewrestrictedmasterproblemS1has10+3=13columns(schedules)andS2has9+3=12columns.Solvingthenewrestrictedmasterproblemvialinearprogrammingyieldsthesolutionx41=x61=x71=x81=0.5andx12=1andx42=x102=0.5.Again,oneofthenarrowbodiesiskeptidle.Thevalueoftheobjectivefunctionis11,575(which,ofcourse,hastobeatleastashighasthevalue11,267ofthefirstlinearprogram,sincenowalargernumberofschedulesistakenintoconsideration).Imposingtheintegralityconstraintsonthismathematicalprogramandsolvingthisrestrictedmasterproblemasanintegerprogramviabranch-and-bound(seeFigure11.5.)yieldsthesolutionx101=x111=1andx12=x22=1.Thevalueoftheobjectivefunctionis11,550. 27411Planning,Scheduling,andTimetablinginTransportation4678111114112221211,211575xxxxxxxUB========361011111114112221211,211575xxxxxxxUB========10111211221,111550Integer optimal solutionxxxxUB=====47811116722211,211,211175xxxxxxUB=======1611411221111525xxxxUB=====47111Infeasiblexx==141167221110725xxxxUB=====410x=411x=610x=611x=710x=711x=Fig.11.5.Optimalsolutionforaircraftschedulingproblem11.4TrainTimetablingThemostcommontraintimetablingproblemfocusesonasingle,onewaytrackthatlinkstwomajorstationswithanumberofsmallerstationsinbe-tween.Atrainmayormaynotstopatasmallerstation.Thiscaseisofinterestbecauserailwaynetworksusuallycontainanumberofimportantlines,referredtoascorridors,thatconnectmajorstations.Thesecorridorsaremadeupoftwoindependentone-waytracksthatcarrytrafficinoppositedirections.Oncethetimetablesforthetrainsinthecorridorshavebeendetermined,itisrela-tivelyeasytofindasuitabletimetableforthetrainsontheotherlinesinthenetwork.Thetimeistypicallymeasuredinminutes,from1toq,whereqrepresentsthelengthofthegivenperiod,e.g.,from1to1440whentheperiodisoneday(similartotheairlineindustry).Linkjconnectsstationj−1withstationj.ThereareLconsecutivelinks(numbered1toL)andthereareL+1stations(numbered0toL).Stations0andLarethefirstandlaststation.LetTdenotethesetoftrainsthatarecandidatestoruneveryperiod.LetTj 11.4TrainTimetabling275denotethesetoftrainsthatintendtopassthroughlinkj.ThesetTjmaybeasubsetofTbecauseatrainmaystartoutfromanintermediatestationand/orendupatanintermediatestation.Trackcapacityconstraintsensurethatonetraincannotpassanotheronthesingletrackbetweentwostations.Atraincanovertakeanotheronlyatastation,whenthetrainthatisbeingovertakenmakesastop.Trainschedulesareusuallydepictedinaso-calledtime-spacediagram(seeFigures11.6and11.7).Suchadiagramenablestheusertodetectconflicts,suchasinsufficientheadways.Foreachtrainthereisanidealtimetable,whichisthemostdesirabletimetableforthattrain.Thisidealtimetableisdeterminedbyanalyzingpas-sengerbehaviorandpreferences.However,thistimetablemaybemodifiedinordertosatisfytrackcapacityconstraints.Itispossibletoslowdownatrainand/orincreaseitsstopping(dwelling)timeatastation.Moreover,onecanmodifythedeparturetimeofeachtrainfromitsfirststation,orevencancelatrain.Thefinalsolutionofthetimetablingproblemisreferredtoastheactualtimetable.Anactualtimetablespecifiesfortraini,i∈T,itsdeparturetimefromitsfirststationanditsarrivaltimeatitslaststation.Thetimetableofeachtrainisperiodic,i.e.,itiskeptunchangedeveryperiod.Thetraintimetablingprobleminvolvesdeterminingthevaluesofvarioussetsofvariables,namelythetimesofarrivalsanddeparturesoftrainiatallstations.Inthissectionthefollowingnotationisused:the(continuous)decisionvariableyij=thetimetrainienterslinkj(i.e.,thetimetrainidepartsfromstationj−1);zij=thetimetrainiexitslinkj(i.e.,thetimetrainiarrivesatstationj).Whenatimetableisputtogetherthereareusuallysomepredeterminedarrivalanddeparturetimesforcertaintrainsatspecificstationsandsomepreferredarrivalanddeparturetimesforothertrains.Thereisacost(orrev-enueloss)associatedwithdeviatingfromthesepreferredarrivalanddeparturetimes.Ifzijisthearrivaltimeoftrainiatstationj,thenthecostfunctionthatspecifiestherevenuelossduetoadeviationfromthispreferredarrivaltimeisdenotedbycaij(zij).Thisfunctionmaybepiece-wiselinearorconvexwithaminimumatthemostpreferreddeparturetime,seeFigure11.8.Apiece-wiselinearfunctionhascomputationaladvantages,sincetheoptimizationproblemcanbehandledinawaythattakesadvantageofthelinearity.Similarly,therearecostsassociatedwiththedeparturetimeoftrainifromstationj,i.e.,yi,j+1,thetravel(trip)timeoftrainionlinkj,denotedbyτij=zij−yijj=1,...,L,andthestopping(dwelling)timeatstationj,denotedbyδij=yi,j+1−zijj=1,...,L−1.Therearedeviationcostsassociatedwitheachoneofthesequantities,namelycdij,cτij,cδij,whicharealsopiece-wiselinearandconvex. 27611Planning,Scheduling,andTimetablinginTransportation54321151515303030454545000000Fig.11.6.TrainTime-Distance(Pathing)DiagramsFig.11.7.Traindiagramgraphicaluserinterface 11.4TrainTimetabling277tcostMost preferreddeparture timeFig.11.8.CostasaFunctionofDepartureTimeThegeneralobjectivetobeminimizedinthetraintimetablingproblemisi∈TLj=1caij(zij)+cdi,j−1(yij)+cτij(zij−yij)+i∈TL−1j=1cδ(yi,j+1−zij)Thevariablesaresubjecttovarioussetsofoperationalconstraints.Forexample,trainineedsatleastaminimumtimeτminijtotraverselinkj;trainimuststopatstationjforaminimumamountoftimeδminijtoallowpassengerstoboard.Forreasonsofsafetyandreliabilityminimumheadwayshavetobemaintainedoneachlink.LetHdhijbetheminimumheadwayrequiredbetweenthedeparturesyh,j+1andyi,j+1oftrainshandifromstationjandletHahijdenotetheminimumheadwaysbetweenthearrivalszhjandzijoftrainshandiatstationj(ensuringadequateheadwaybetweentrainshandiwhenexitinglinkj).Moreover,theremaybeupperandlowerboundsonallarrivalanddeparturetimes.FormulatingthetimetablingproblemasaMixedIntegerProgram(MIP)requiresasetof0−1variables:thedecisionvariablexhijassumesthevalue1iftrainhimmediatelyprecedestrainionlinkj,and0otherwise.Also,tomaketheformulationeasier,twodummy(artificial)trainsiandiareincludedinsetT:trainihasfixedarrivalanddeparturetimesensuringthatitprecedeseveryothertrainoneachlinkandtrainialsohasfixedarrivalanddeparturetimesensuringthatitwillbethelasttrainoneachlink.The 27811Planning,Scheduling,andTimetablinginTransportationtimetablingproblemcannowbeformulatedasthefollowingmixedintegerprogram.minimizei∈TLj=1caij(zij)+cdi,j−1(yij)+cτij(zij−yij)+i∈TL−1j=1cδ(yi,j+1−zij)subjecttoyij≥yminiji∈T,j=1,...,Lyij≤ymaxiji∈T,j=1,...,Lzij≥zminiji∈T,j=1,...,Lzij≤zmaxiji∈T,j=1,...,Lzij−yij≥τminiji∈T,j=1,...,Lyi,j+1−zij≥δminiji∈T,j=1,...,L−1yi,j+1−yh,j+1+(1−xhij)M≥Hdhiji∈T,j=0,...,L−1zij−zhj+(1−xhij)M≥Hahiji∈T,j=1,...,Lh∈{T−i}xhij=1i∈T,j=1,...,Lxhij∈{0,1}Mostoftheconstraintsetsareself-explanatory.However,someofthecon-straintsetsmaywarrantsomeexplanation.ThevalueMintheseventhandeighthconstraintsetismadeverylargeforthefollowingreason:iftrainhisnotimmediatelyprecedingtrainionlinkj,thenxhij=0andtheinequalitieswithanMonthelefthandsideareautomaticallysatisfied;iftrainhprecedestrainionlinkj,thenxhij=1andtheheadwayconstraintmustbeenforced.Sincetheintegerprogramaboverepresentsonlyaproblemconcerningasinglelinkandtherailwaysystemtypicallyhastosolveanetworkconsistingofmanylinks,itisimportanttosolvethesinglelinkproblemfast.Oneheuristicisbasedondecomposition.Accordingtothisheuristicthetrainsarescheduledoneatthetimeandasolutionfortheoverallproblemisgeneratedbysolvingaseriesofsingletrainsubproblems.InagiveniterationitisassumedthatinpreviousiterationsthesequenceshavebeendeterminedinwhichtrainsbelongingtosetT0gothroughthedifferentlinks.ThissetT0includesthetwoartificial(dummy)trainsiandiwhicharethefirstandlasttrainoneachlink.ThenextiterationhastoselectatrainfromsetT−T0forinclusioninsetT0anddetermineforeachlinkwherethistrainisinsertedinthesequenceoftrains.ThemannerinwhichthetrainsareselectedforinclusioninsetT0canaffectthequalityofthesolutiongeneratedaswellasthespeedatwhichthesolutionisobtained.Oneverysimplepriorityruleselectsthetrainsintheorderoftheirdesiredstartingtimes.Anotherpossiblepriorityruleselectsthe 11.4TrainTimetabling279trainsindecreasingorderoftheirimportance;theimportanceofaparticulartrainmaybemeasuredbythetrain’stype,speed,andexpectedrevenue.Forexample,adescendingorderofimportancemaybe:expresstrainswithoutstops,expresstrainswithstops,localtrainsandfreighttrains.Athirdrulewouldselectatrainthathaslittleflexibilityinitsarrivalanddeparturetimes.Afourthpriorityrulecanbedesignedbycombiningthefirstthreerules.AssumethatinagiveniterationtrainkisselectedforinclusioninsetT0.So,thetrainsinsetT0havealreadybeenscheduledinthepreviousiteration.Eventhoughthecomputationsinthepreviousiterationdeterminedexactarrivalanddeparturetimesofthetrainsatallstations,theonlyinformationthatiscarriedforwardtothecurrentiterationisthesequenceororderinwhichthetrainsinsetT0traverseeachlink(theexactarrivalanddeparturetimesaredisregarded).LetIjdenotethevectoroftrainindicesthatspecifiestheorderinwhichthetrainsinT0arescheduledtotraverselinkj.ItisassumedthatinthisorderIjtrainiisimmediatelyfollowedbytraini∗.ThecurrentiterationmustscheduletrainkoneachlinkwhilemaintainingtheorderIjinwhichthetrainsinT0traverselinkj,i.e.,trainkisinsertedsomewhereinvectorIj.However,eventhoughtheorderingsIj,j=1,...,L,aremaintained,theexactarrivalanddeparturetimesofthetrainsinT0areallowedtovaryaftertheinclusionofk.ThesubproblemthatinsertstrainkinthevectorsI1,...,ILisamixedintegerprogram,sayMIP(k),whichissimilarbutsimplerthantheMIPformulatedfortheoriginalproblem.InordertosimplifytheformulationofMIP(k),includetrainkinsetT0.MIP(k)insertstrainkineachvectorIjanddeterminestheexactarrivalanddeparturetimesforalltrainsinsetT0.TheobjectivefunctionisthesameastheobjectivefunctionoftheMIPfortheoriginalproblem.However,theconstraintsetshavetobemodified.Adifferent0−1variableisusedinMIP(k),namelyxijwhichis1iftrainkisinsertedonlinkjimmediatelyaftertrainiand0otherwise.Thatis,xijis1iftrainkisscheduledinbetweentrainiandthetrainimmediatelyfollowingtraini,whichinvectorIjisreferredtoastraini∗.LetsetT0nowincludealltrainsalreadyscheduledinthepreviousiterationsaswellastraink.TheintegerprogramMIP(k)canbeformulatedasfollows:minimizei∈T0Lj=1caij(zij)+cdi,j−1(yij)+cτij(zij−yij)+i∈T0L−1j=1cδ(yi,j+1−zij)subjecttoyij≥yminiji∈T0,j=1,...,Lyij≤ymaxiji∈T0,j=1,...,Lzij≥zminiji∈T0,j=1,...,Lzij≤zmaxiji∈T0,j=1,...,L 28011Planning,Scheduling,andTimetablinginTransportationzij−yij≥τminiji∈T0,j=1,...,Lyi,j+1−zij≥δminiji∈T0,j=1,...,L−1yi∗,j+1−yi,j+1≥Hdii∗ji∈Ij,i=i,j=0,...,L−1yk,j+1−yi,j+1+(1−xij)M≥Hdikji∈Ij,i=i,j=0,...,L−1yi∗,j+1−yk,j+1+(1−xij)M≥Hdki∗ji∈Ij,i=i,j=0,...,L−1zi∗j−zij≥Haii∗ji∈Ij,i=i,j=1,...,Lzkj−zij+(1−xij)M≥Haikji∈Ij,i=i,j=1,...,Lzi∗j−zkj+(1−xij)M≥Haki∗ji∈Ij,i=i,j=1,...,Li∈{T0−k−i}xij=1j=1,...,Lxij∈{0,1}ThusMIP(k)hasthesamecontinuousvariablesyijandzijastheoriginalMIP,butcontainsfarfewer0−1variablesthantheoriginalMIP.Nowthebinary0−1variablesarexijratherthanxhij.BeforesolvingMIP(k)itisadvantageoustodosomepreprocessinginordertoreducethenumberof0−1variablesandthenumberofconstraints.Manyofthe0−1variablesxijarelikelytoberedundant,duetoupperandlowerboundsinotherconstraintsonthedeparturetimes,arrivaltimes,stoppingtimes,andoveralltriptimes.Forexample,thevariablexhjisnotneedediftrainkcannotbescheduledimmediatelyaftertrainhonlinkj(i.e.,inbetweentrainshandh∗),becauseofalackofsufficienttimeinbetweentrainshandh∗(notallowingfortherequiredheadways).SolvingMIP(k)canbedonethroughbranch-and-bound.Thereareseveralstrategiesonecanfollowinthebranchingprocess.Atagivennodeinthebranchingtree,allbranchescorrespondtoacertainlinkj.Morespecifically,thereisabranchforeachmemberofthesetoftrainsalreadyscheduledthattrainkwouldbeabletofollow.Thismultiplebranching(branchingdependentonthetrainsalreadyscheduled)canbecombinedwithfixedorderbranchingthatisbasedonthelinks.Thatis,thelinksareintheorderinwhichtheyaretraversedbythetrains.Summarizing,thefollowingheuristicframeworkcanbeusedtofindagoodfeasible(oroptimal)solutionfortheoriginalMIP.Algorithm11.4.1(TrainTimetabling).Step1.(Initialization)Introducetwo“dummytrains”asthefirstandlasttrains.Step2.(SelectanUnscheduledTrainandSolveitsPathingProblem)Selectthenexttrainkthroughthetrainselectionpriorityrule.Step3.(SetupandPreprocessMixedIntegerProgram)IncludetrainkinsetT0. 11.5JeppesenSystems:DesignandImplementation281SetupMIP(k)fortheselectedtraink.PreprocessMIP(k)toreducenumberof0−1variablesandconstraints.Step4.(SolveMixedIntegerProgram)SolveMIP(k).Ifalgorithmdoesnotyieldfeasiblesolution,STOP.Otherwise,addtrainktothelistofalreadyscheduledtrainsandfixforeachlinkthesequencesofalltrainsinT0.Step5.(RescheduleAllTrainsScheduledEarlier)Considerthecurrentpartialschedulethatincludestraink.Foreachtraini∈{T0−k}deleteitandrescheduleit.Step6.(StoppingCriterion)IfT0consistsofalltrains,thenSTOP;otherwisegotoStep2.Ofcourse,theframeworkpresentedaboveismerelyaheuristic.Itdoesnotguaranteetoyieldanoptimalsolution.Thefollowingexampleillustrateshowtheheuristicmayresultinasuboptimalsolution.Example11.4.2.Considerfourtrainsthathavetobescheduledonasinglelink.Twotrains(AandD)arefasttrainswithtraveltimesτ1andtwotrains(BandC)areslowtrainswithtraveltimesτ2.TrainAmustdepartatafixedtime,say9a.m.,andtheotherthreetrainshavetoeitherarriveordepartatatimethatisascloseto9a.m.aspossible.IfthetrainsareintroducedoneatatimeintheorderA,B,C,D,thentheresultingscheduleisasdepictedinFigure11.9(a).Thissolutionisalocaloptimum.TheglobaloptimumisasdepictedinFigure11.9(d).ThedecompositionframeworkdescribedaboveisinsomerespectsverysimilartotheshiftingbottleneckheuristicdescribedinChapter5forjobshopscheduling.First,thetraintimetablingframeworkschedulesonetrainatthetime.Ineachiteration,thearrivaltimesanddeparturetimesofallthetrainsinsetT0arecomputed;however,thearrivaltimesanddeparturetimesaresubjecttomodificationinsubsequentiterations.Third,afterschedulinginaniterationoneadditionaltrain,itreschedulesallthetrainsthathadbeenscheduledinthepreviousiterations.11.5JeppesenSystems:DesignandImplementationSeveralsoftwarecompaniesspecializeinplanningandschedulingsystemsforthetransportationindustries.JeppesenSystems,aBoeingcompany,isoneofthemoresuccesfulcompaniesinthisdomain.Jeppesenacquiredavery 28211Planning,Scheduling,andTimetablinginTransportationABCDABCDABCDABCD(a) A locally optimal solution of P.(c) A suboptimal solution of P.(d) A global optimal solution of P.(b) A suboptimal solution of P.9.009.009.009.00Fig.11.9.TrainSchedulesforExample11.4.2successfulSwedishsoftwaremakercalledCarmenSystemsin2006andincor-poratedtheCarmensoftwareinitsproductline-up.JeppesenSystemsnowoffersanextensivesuiteofproductsfortheairlineindustry(seeFigure11.10).Jeppesen’sproductlineisbasedonCarmen’sproprietarymodellinglan-guageRave.Thismodellinglanguagemakesiteasyfortheusertodescribeandimplementcostfunctions,feasibilityconditions,andqualityconstraints.ThislanguagesupportsasuiteofJeppesen’sproductsandisessentialforthefinetuningofeverydayschedulesandforconductingsimulations.Thelanguageisdesignedinsuchawaythataprogramcanbelinkedeasilytoadvancedoptimizationroutines.AplannerwhousesasystembasedonRavecanreadilyimplementmodificationsintheoperationprinciplesoftheairlinebysimplyswitchingaruleinthecodeonoroff,orbyadjustingthevalueofaparameterinarule.Example11.5.1.Aruleensuringthatitisimpossibletohaveacrewassign-mentlongerthan10hourscanbewrittenasfollows:rulemax-duty=%duty%<=10:00;remark”Maximalduty”;endThefollowingisanexampleofacostfunctionthatconsistsofcomponentsforsalary,hotel,perdiemandflightpositioning:%pairingcost%=%salary%+%hotel%+%perdiem%+%positioning%+.... 11.5JeppesenSystems:DesignandImplementation283Carmen Rave (General Modelling Language)Carmen Roster MaintenanceCarmen Crew PairingCarmen Crew RosteringCarmen Fleet ControlCarmen Fleet AssignmentCarmen Tail AssignmentCarmen PassengerRecoveryCarmen Crew ControlCarmen ManpowerCarmen PassengerRecoveryCarmenTail AssignmentCarmen FleetAssignmentCarmenManpowerCarmen CrewPairingCarmen CrewRosteringCarmen RosterMaintenanceCarmen CrewControlCarmen FleetControlCarmen Rave (General Modelling Language)Carmen IntegratedOperation ControlProduct OverviewFig.11.10.JeppesensystemsproductoverviewJeppesen’sclientstypicallyhavetheirownruleprogrammerswhoensurethatJeppesen’sproductskeeponsolvingtherightproblemsinachangingenvironment.Carmen’sRavealsoenablesuserstoperformsimulationstudiesandconsiderwhat-ifscenarios.Therulesthatareembeddedinonemoduleareautomaticallyfollowedinanothermodule.Forexample,crewconnectionrulesestablishedinCarmenCrewPairingareautomaticallyadheredtoinCarmenTailAssignment.TwoofthemoreimportantmodulesintheJeppesenproductsuitearetheFleetAssignmentmoduleandtheTailAssignmentmodule.FleetAssignmentistheprocessofassigningfleetsoraircrafttypestotheflightsintheschedule;thismoduleismainlyconcernedwithgeneratingaschedulethatisbalanced(e.g.,atanyairport,thenumberofarrivalsofaircraftofanygiventypehastobeequaltothenumberofdeparturesofaircraftofthattype),butisnotconcernedwithindividualaircraftandtheiroperationalconstraints,suchasmaintenance.TheTailAssignment,ontheotherhand,istheproblemofdecidingwhichindividualaircraft(specifiedbyitstailnumber)shouldcoverwhichflight.Themainfocusofthetailassignmentistoverifywhetherthescheduleis 28411Planning,Scheduling,andTimetablinginTransportationfeasible;ithastodealthereforewithindividualaircraftandtakeoperationalconstraintsintoaccount.Suchoperationalconstraintsinclude:(i)minimumconnectiontimesandmaintenanceconstraints;(ii)routeconstraints,e.g.,onlyaircraftequippedtoflyoverseascanbeassignedtosuchflights;(iii)thatonlyaircraftwith”hushkits”canbesenttonoise-restrictedair-ports;(iv)destinationconstraints,e.g.,onlyaircraftthatdonotrequireanex-ternalpowersupplycanbesenttoairportswheresuchaserviceisunavailable.TheTailAssignmentalsoconstrainsthecrewplanning.Ifacrewremainswiththesameaircraftontwoconsecutivelegs,thentheconnectiontimebetweenthetwoconsecutiveflightsinacrewrostercanbeshorterthanifthecrewhadtochangeaircraft.Byminimizingtheconnectiontimesinacrewroster,aschedulecanbecreatedwithlesscrew.Furthermore,acrewremainingwithanaircraftmakestheschedulemorerobustandlessvulnerabletodisturbancesthatoccurduringthedayofoperation.ByallowingthecrewplanningtoinfluencetheTailAssignment,itispossibletoreducecostsandincreasestabilityatthesametime.Thisimpliesanadditionaloperationalconstraint:(v)fixedlinksbetweenflightstoprotectcrewrosters.Theseconstraintscanalsobemodeledascostsincurredwhenlinksarebroken.TheTailAssignmentusuallytakestheFleetAssignmentasgiven.How-ever,manyairlinesarenowlookingintothepossibilityofbeingmoreadaptivetoactualbookinglevelsandchangethefleetassignmentclosetotheday-of-operationinordertoaccomodatemorepassengers.ThisapproachisreferredtoasDemandDrivenDispatch.ThischangestheformulationoftheaircraftroutingandschedulingproblemdescribedinSection11.3fromacostmini-mizationproblemtoaprofitmaximizationproblem.Arevenuecomponenthastobeaddedtotheobjectivefunctioninordertomakesurethattherevenuedoesdependonthetypeofaircraftthatisassignedtotheflight.Insteadofminimizingcostscli,onehastomaximizeprofitsπli;theprofitπliisequaltotherevenuegeneratedwiththeassignmentofaircraftitoschedulelminuscli.Inpractice,theproblemformulationhasmoreconstraintsthantheformulationinSection11.3;thereareusuallyadditionalconstraintsthatensurethattheaircraftassignmentsarefeasiblegiventhealreadyestablishedcrewschedules(thecrewschedulesareoftengeneratedbeforetheaircraftassignmentsaremade).InthepureTailAssignmentapplication,theoptimizerisusedtogener-ateaircraftroutesthatautomaticallysatisfyallfeasibilityconstraints(withregardtominimumturn-aroundtimes,maintenancetimes,andsoon).Theobjectiveincludesmaximizingaircraftutilization,prioritizingrobustconnec-tionswithasfewcriticalaircraftchangesaspossible(i.e.,maximizingschedulestabilityandrobustness).Theoutputoftheoptimizationmoduleisthemost 11.6Discussion285efficientcombinationofaircraftrotations.Byusingtheaircraftefficientlyitispossibletofreeupmoreaircraftforstand-byduty.Anaircraftbeingidlefor2-3hoursmaybeawasteofaresource,whereasanaircraftbeingonthegroundfor6hoursormoremaybeequivalenttoaplanebeingavailableforstand-byduty.Bypromoting,atthesametime,veryshortandverylongcon-nections,thenumberofplanesavailableforstand-bydutycanbeincreasedconsiderably.WhenthesystemisusedaccordingtoDemandDrivenDispatch,revenueinformationisusedinordertodetermineifitisadvantageoustochangeair-crafttypes.Mostmajorairlineshavecrewsthatcanoperatedifferentaircrafttypeswithvaryingseatcapacity.Thisenablestheairlinetomakelateadjust-mentstothefleetassignmentswithouthavingtochangethecrewrosters.Theoptimizerallocatestheaircraftinthemostefficientway,balancingtherev-enuesandcosts.Theuser(i.e.,thescheduler)determinestheflightsforwhichitispermittedtochangethefleetassignment.Theoptimizerthensolvestheproblemfortheindividualtailswithprofitmaximizationasobjective.Also,becauseofRave’ssimulationcapabilitiesitispossibletomakemuchmoreinformeddecisionswhennewschedulesarebeingcreated.Thegraphicaluserinterface(GUI)enablestheplannertoaddormodifyrelevantinformation.TheGUIalsoprovidesanoverviewofsolutionsandinputdataintheformofGanttcharts,seeFigure11.11.ThroughtheGUItheusercanselectflightsoronlyasubsetoftheplantomodifyandimprove.Heorshecaninsertmaintenanceactivitiesforspecificplanesintheschedule,andsoon.Thereportgeneratormakesitpossibletoissuereportsatanytime(e.g.,maintenancereports,airportstand-bys,andcrewstabilityanalysis).Reportscanbedisplayedonscreenoronpaper.Thereportgeneratorcanalsoexportdatafilestootherdecisionsupportsystems.Jeppesenhasinstalledsystemsfornumerousairlines,includingBritishAirways,NorthwestAirlines,SingaporeAirlines,andAeroMexico.Whenasystemisintroduced,savingsintheorderof5%aretypicallyachievedim-mediately.However,airlinesexpectthelongtermsavingstobesignificantlyhigher.11.6DiscussionThethreetransportationsettingsconsideredinthischapterhavesimilaritiesaswellasdifferences.Allthreeproblemsleadtointegerprogrammingfor-mulations.Inonecasethemodelresultsinasetpackingformulationandinanothercasethemodelresultsinasetpartitioningformulation.Therearevarioussimilaritiesbetweenaircraftschedulingandrailwaytimetabling.Bothaircraftschedulingandrailwaytimetablingrequirecyclic(periodic)schedules;theperiodsaretypicallyoneday(withinternationalflightstheperiodsmaybeoneweek).Inbothaircraftschedulingandrailway 28611Planning,Scheduling,andTimetablinginTransportationFig.11.11.JeppesenSystemsaircraftschedulinggraphicaluserinterfacetimetablestheinputrequirementsandtheconstraintsontheschedulesaredeterminedinadvancebymarketingdepartments.Theanalysisofcustomerbehaviorandpreferencesdetermine,withintimeranges,theidealdeparturetimesoftheplanesandtrains.Inrailwaytimetablingaswellasinaircraftschedulingweekenddaysaredifferentfromweekdays.Thereareofcoursealsosomedifferencesbetweenaircraftschedulingandplaneinthesamewayasalinkbetweentwostationshastobecoveredbyatrain.However,flightlegshavetobecombinedwithoneanotherintoroundtrips,whereastrainshavetomoveinmajorcorridors.Therearemanydifferencesbetween,ontheonehand,tankerschedulingproblemsand,ontheotherhand,aircraftschedulingandtraintimetablingproblems.Tankerandshippingschedulesaretypicallynotcyclic.Schedulesaregeneratedonarollinghorizonbasis,planningaheadforthreemonths.Ittendstobelesscomplexthantheaircraftschedulingproblemandtherailwaytimetablingproblem.(Thesourcesofcomplexitiesinaircraftschedulingarebasedonthefactthatflightlegshavetobecombinedintoroundtrips,andrailwaytimetabling.Aflightlegbetweentwoairportshastobecoveredbya 11.6Discussion287inrailwaytimetablingthattrackshavecapacityconstraintsandbypassingisonlyallowedatstations).Anotherimportantdifferencebetweentankerschedulingandaircraftschedulingisbasedonthefactthatintankerschedulingeachshiphasitsownidentity.Inaircraftscheduling,adistinctionismadebetweenaircrafttypes;aircraftthatareofthesametypeareinterchangeable.Itisimportantforanairlinenottohavetomakeadistinctionbetweenaircraftofthesametype.Thisway,thesizeoftheintegerprogramcanbereducedconsiderably.Anotherdifferencebetweenaircraftschedulingandtankerschedulingliesintheobjectivefunction.Theobjectivefunctioninthetankerschedulingproblemtakesintoaccountthatcertaincargoesmaybecarriedbyshipsthatarenotcompany-owned.Thenecessarypaymentsaredeterminedonthespotmarket.Anycargothatistransportedonacompanyshiprepresentsasavingorprofit.Therearemanysimilaritiesbetweentransportationschedulingmodelsandmachineschedulingmodelsinmanufacturing:Thetankerschedulingproblemandtheaircraftroutingandschedulingproblemaresomewhatsimilartoparal-lelmachineschedulingproblems.Tankersandplanescorrespondtomachines,whilecargoesandflightlegscorrespondtojobs.Afleetoftankersoraircraftmaythereforebeconsideredequivalenttoaparallelmachineenvironment.Transportingacargoorcoveringalegissimilartoprocessingajob.Thejobshavecertainprocessingtimesandalsotimewindowsduringwhichtheyhavetobeprocessed.Whenatankertransportsonecargofollowedbyanother,acertaintypeofsequencedependentsetuptimeandsetupcostisincurred.Thesameistruewhenagivenplanefliesonelegfollowedbyanother.Theobjectivesare,ofcourse,veryproblemspecific.Intheaircraftroutingproblemtheobjectivesareafunctionofthesumofearlinessandthesumoftardinesscosts.Ifonecomparestraintimetablingwithmachinescheduling,thenthelinksmustrepresentthemachines,andthetrainsthejobs.Thetraintimetablingproblemisthensomewhatsimilartoaflowshopproblem,i.e.,anumberofmachines(links)havetoprocessasetofjobs(trains)oneafteranother.Eachjobhastobeprocessedfirstonmachine1(link1),thenonmachine2,andsoon.Notethatthisisdifferentfromapermutationflowshopinwhicheachmachineprocessesallthejobsinthesamesequence.Onemachinemayprocessthejobsinadifferentsequencethananothermachine,becauseinthetrainschedulingproblem,itmayoccurthatatastationonetrainpassesanother.However,thereisanotherimportantdifferencebetweenthetrainschedulingproblemandtheflowshopproblem.Inaflowshopproblem,amachineisnotallowedtoprocessmorethanonejobatthesametime.Inthetrainschedulingproblem,alinkmaybe”processing”morethanonetrainatatime(seeExercise11.8).Inthischapterwedidnotcovertheroutingandschedulingoftrucks.However,theroutingandschedulingoftrucksandtankertrucks(whichare 28811Planning,Scheduling,andTimetablinginTransportationoftenreferredtoasroadtankers)issomewhatsimilartotheroutingandschedulingoftheoiltankersdescribedinthesecondsection.Inmanytransportationsettingsthereareotherimportantschedulingprob-lemsthathavetobedealtwithaswell.Forexample,aircraftschedulingaffectscrewscheduling.ThecrewofanaircraftisaveryimportantcostcomponentthathastobeminimizedsubjecttounionaswellasFAAconstraints.Othertypesofschedulingproblemsthathavetobedealtwithintheaviationindus-tryincludegatescheduling(i.e.,assigninggatestoincomingaircraft)aswellaspersonnelschedulingforcheck-incounters.Exercises11.1.ConsiderthetankerschedulingprobleminExample11.2.1.Supposethatcl2,foralllismultipliedbyafactorK,K<1.AtwhichvalueofKwouldtanker2beused?11.2.ConsideragainthetankerschedulingprobleminExample11.2.1.Sup-posethatthechartercostofcargo6is2217K,whereKisafactorgreaterthan1.AtwhatvalueofofKdoesitmakesensetotransportcargo6withacompany-ownedship?11.3.Inthesubproblemoftheaircraftroutingandschedulingproblem,thedualvariablesrepresentacertainprofit.Section11.3providesacleardescrip-tionofwhatthevariableβirepresents.Giveadescriptionofwhatthevariablesαjandγiprepresent.11.4.ConsiderExample11.3.1.Thesetofdualvariablesinthefirstlinearprogrammingsolutionoftherestrictedmasterproblemarenotnecessarilyunique.Anothersetofdualvariablesarethefollowing:Legj123456789101112αj0−616.7000183.329502766.7001683.34333.3Thedualvariablescorrespondingtothesecondsetofconstraintsareβ1=−16.67andβ2=0.Thedualvariablescorrespondingtothethirdsetofconstraintsare:airporth1234γ1h0−15000−3550γ2h0−1466.70−3733.3Computethepotentialprofitofschedulesa111j,...,a151janda102j,...,a162j.WhichschedulesshouldbeincludednowinS1andS2?11.5.InExample11.3.1allflightshavetobecoveredbyaplane.Supposetheprofitonleg2isreduced,i.e.,bothπ12andπ22aremultipliedwithafactorK<1.AtwhatlevelofKdoesitmakesensenottocoverleg2? CommentsandReferences28911.6.RedoallthestepsinExample11.3.1assumingthatplanesoftype1arenotallowedtoflyaccordingtoschedule8inS1.11.7.ConsidertheMosel(DashOptimization)codeofExample11.3.1thatisstoredinSection4oftheCD-ROM.Modifyeitherthecodeorthedatafilesinsuchawaythatanemptyscheduleisnotavailableforplanesoftype2(i.e.,setx12equalto0).Comparetheoptimalsolutionwiththeoptimalsolutionpresentedinthetext.11.8.Describethesimilaritiesandthedifferencesbetweenaflexibleflowshop(seeSection2.2)andthetraintimetablingproblem.11.9.ConsiderthefollowingvariationofthetraintimetablingprobleminSection11.4.Insteadoftwotracksbetweenanytwostationsthereisnowasingletrackbetweenanytwostations.Thesingletrackhastoaccomodatethetrafficinbothdirections.Atthestationstrainscanbypassoneanotherinbothdirections.DevelopaheuristicthatgeneratesthetimetableswiththesameobjectivesasdescribedinSection11.4.11.10.ConsiderthetraintimetablingproblemdescribedinSection11.4.ComparethenumberofvariablesandthenumberofconstraintsintheMIPwiththenumbersinMIP(k).CommentsandReferencesNoframeworkhasyetbeenestablishedforplanningandschedulingmodelsintrans-portation.Aseriesofconferencesoncomputer-aidedschedulingofpublictransporthasresultedinanumberofveryinterestingproceedings,seeVossandDaduna(2001),Wilson(1999),Daduna,Branco,andPintoPaixao(1995),DesrochersandRousseau(1992),andWrenandDaduna(1988).AvolumeeditedbyYu(1998)con-sidersoperationsresearchapplicationsintheairlineindustry;thisvolumecontainsseveralpapersonplanningandschedulingintheairlineindustry.Barnhart,Belob-abaandOdoni(2003)presentasurveyofoperationsresearchapplicationsintheairtransportindustry.TheoiltankerschedulingproblemdescribedinthesecondsectionisbasedonthepaperbyFisherandRosenwein(1989)andPerakisandBremer(1992).AdditionaldetailsconcerningExample11.2.1canbefoundontheCD-ROMaccompanyingthistext.Foraslightlydifferentmodelforoceantransportationofcrudeoil,seeBrown,GravesandRonen(1987)andforanothershiproutingmodel,seeChristiansen(1999).Fortwoexcellentoverviewsonroutingandschedulinginthemaritimein-dustry,seeChristiansen,FagerholtandRonen(2004)andChristiansen,Fagerholt,NygreenandRonen(2006).ThedailyaircraftroutingandschedulingformulationisfromDesaulniers,Desrosiers,Dumas,Solomon,andSoumis(1997).MoredetailswithregardtoEx-ample11.3.1canbefoundontheCDthataccompaniesthisbook.Desrosiers,Lasry,McInnes,Solomon,andSoumis(2000)developedaroutingandschedulingsystemcalledALTITUDEandimplementeditattheCanadianairlineTransat.Stojkovic,Soumis,Desrosiers,andSolomon(2002)presentamodelforarealtime 29011Planning,Scheduling,andTimetablinginTransportationflightschedulingproblem.KeskinocakandTayur(1998)andMartin,JonesandKe-skinocak(2003)considertheschedulingoftime-sharedjetaircraft.Formodelsthatintegratefleetassignmentwithcrewpairing,seeBarnhart,LuandShenoi(1998)andCordeau,Stojkovic,Soumis,andDesrosiers(2001).ThetraintimetablingapproachdescribedinthischapterisduetoCareyandLockwood(1995).Foranother,moreelaborateapproach,seeCaprara,FischettiandToth(2002). Chapter12PlanningandSchedulinginHealthCare12.1Introduction.................................29112.2SchedulingaSingleOperatingRoom...........29212.3MultipleOperatingRooms-ASetPackingFormulation..................................29712.4MultipleOperatingRooms-AStochasticApproach....................................30112.5PlanningandSchedulingRadiotherapyTreatments..................................30412.6EmergencyRoomStaffing-AConstraintProgrammingApproach.......................30812.7ASurgerySchedulingandBedOccupancyLevellingSystem.............................31012.8Discussion...................................31312.1IntroductionInmostcountriesthehealthcareindustryrepresentsasignificantpercentageoftheGrossNationalProduct.Withtheagingofthepopulation,healthcarecostshavebeengoingupconsiderablyoverthelastcoupleofdecades.Forthesereasons,manygovernmentshavebeguntolookintotheproductivityoftheirhealthcareindustries.Healthcareproductivityistoagreatextentbasedontheproperplanningandschedulingofalltheactivitiesinvolved.Thevarietyinplanningandschedulingprocessesinhealthcareturnsouttobeimmense.Thischapterfocusesonjustafewoftheseprocesses.Themostexpensiveresourcesinahospitalaretypicallytheoperatingroom(s),thesurgeons,andtheanesthesiologists.Anhourofoperatingroomtimeisveryexpensiveandsoisanhourofasurgeon’stime.Theoperating© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,291DOI: 10.1007/978-1-4419-0910-7_12, 29212PlanningandSchedulinginHealthCareroomsinahospitalcanactuallyberegardedasthehospital’sengine.Thesurgeriesdrivethedemandformanyofthehospital’sdivisions,suchastheIntensiveCareUnits(ICUs)andotherpostoperativecareunits(includingbeds,nurses,andsoon).Commonobjectivesinanoperatingtheatrearetheminimizationoftheidletimesoftheoperatingroomsandtheminimizationofthetimessurgeonshavetowaitforanoperatingroomtobecomeavailable.Sincethedurationsofsurgeriesaretypicallyrandom,theplanningandschedulingofthesurgeriesisusuallynotasimpletask.Theschedulingofanoperatingtheatre(consistingofanumberofoperatingrooms)isverysimilartotheparallelmachineschedulingproblemdescribedinChapter5sinceanoperatingroommayberegardedasamachineandasurgerymayberegardedasajob.Theschedulingofanoperatingroommayalsoberegardedasatimetablingproblem(seeChapter9).Thefirstthreesectionsofthischapterdescribeseveralapproachesfordealingwiththeschedulingofoperatingrooms.Thischapteralsoconsidersotherplanningandschedulingproblemsthatareofimportanceinhealthcare.Forexample,theplanningandschedulingofradiotherapytreatments(whichisaspecialcaseofanappointmentschedulingproblem).Anotherexampleofaplanningandschedulingprobleminahealthcaresettingthatisconsideredinthischapteristheassignmentofphysicianstoemergencyroomshifts.Thisproblemisaspecialcaseofaworkforceschedulingproblem.(Workforceschedulingandstaffingwillbeconsideredinmoredetailinthenextchapter.)ThelastsectionofthischapterdescribesanactualsystemforsurgeryschedulingandbedoccupancylevellinginahospitalinBelgium.12.2SchedulingaSingleOperatingRoomThebasicproblemconcerningsurgeryschedulingistheunderlyingrandom-ness.Thedurationofanoperationisinherentlyrandom.Thedistributionisoftenknownsincemosthospitalskeepdetailedstatisticswithregardtothesurgeries.Theschedulerofthesurgeriesinahospitalfacesactuallytwoproblems.First,hehastodetermineinwhichsequencetheoperationsshouldbeper-formed.Second,giventhesequence,hehastotellthesurgeonwhowillperformthekthoperationinthesequencethetimeatwhichtheoperatingroomwillbemadeavailabletohim.Intheanalysisthatfollows,thesubscriptkreferstooperationk,whereasthesubscript(k)referstothekthoperationinagivensequence.Sotheoper-ationreferredtobysubscript(k)doesnotrefertooperationksinceoperationkmaybeassignedtoadifferentpositioninthesequence.Assumethat,givenasequenceofoperations,thesurgeonwhohastoper-formthe(k+1)stoperationinthesequenceistoldthattheoperatingroomwillbeavailableattimed(k)(i.e.,d(k)mayberegardedasaformofduedate 12.2SchedulingaSingleOperatingRoom293forthekthoperationonthelist).Ifitturnsoutthatthekthoperationiscom-pletedearlyatC(k)d(k),thenthesur-geonhastowaitfortheoperatingroomtobecomeavailable.Thewaitingtimeofasurgeoninvolvesacostwsg(C(k)−d(k)).Ifasequenceofnoperationsisgiven,theschedulerhastodeterminethetimesd(1),d(2),…,d(n−1)insuchawaythatthetotalcost(operatingroomaswellassurgeoncost)isminimized.Someadditionalnotationisneeded.LettherandomvariableYjdenotethedurationofoperationj,letfjdenoteitsdensityfunctionandFjitsdistributionfunction.Considerfirstasimplecasewithtwosurgeries.Ifthesurgeriesarescheduledinthesequence1,2andthesurgeonperformingthesecondoperationispromisedtheoperatingroombytimed(1),thenthetotalexpectedcost,asafunctionofd(1),isd(1)0wor(d(1)−x)f1(x)dx+∞d(1)wsg(x−d(1))f1(x)dx.Theredoesnotexistaclosedformexpressionfortheoptimalvalueofd(1)foranarbitrarydensityfunctionf1.However,theoptimalvalueofd(1)hasthefollowingproperty:Themarginalcostofpostponingd(1)byasmallamountoftimeδmustbeapproximatelyequaltothemarginalbenefitofsuchapostponement.TheexpectedmarginalcostistheprobabilitythattherandomvariableY1islessthand(1)multipliedbythecostofkeepingtheoperatingroomempty,i.e.,worF1(d(1)).TheexpectedmarginalbenefitistheprobabilitythattherandomvariableY1islargerthand(1)timesthewaitingcostofthesurgeon,i.e.,wsg(1−F1(d(1))).(Thereisabenefitnow,becausethesurgeonhaslessofawait.)Settingthemarginalcostequaltothemarginalbenefityieldsanequationthattheoptimalvalueofd(1)mustsatisfy:F1(d(1))=wsgwsg+woror,equivalently,usingtheinversefunctionofthedistributionfunctionF1,d(1)=F−11wsgwsg+wor.Assumenowthattheschedulerhastodecidetheoptimalsequenceofthetwosurgeries.Whichoneofthetwosequencesisbetter:1,2or2,1?(Assumingthattheschedulerdeterminesforbothsequencestheoptimaltimeatwhichtheoperatingroomispromisedtothesecondsurgeon(d(1)).)Example12.2.1(SchedulingTwoConsecutiveSurgeries).ConsidertwosurgerieswithrandomdurationsY1andY2.TherandomvariableY1isUniformlydistributedoverthesupport[9,21],i.e.,E(Y1)=15andVar(Y1)=12.TherandomvariableY2isuniformlydistributedoverthesupport[37,43],
29412PlanningandSchedulinginHealthCarei.e.,itsmeanE(Y2)=40anditsvarianceVar(Y2)=3.Assumewor=1andwsg=2.Considerfirstsequence1,2.Theoptimalscheduledstartingtimeofthesecondoperationwouldbed(1)=F−11(2/3)=17andthetotalexpectedcostunderthisschedulewouldbe179(17−x)112dx+21172(x−17)112dx=2.6666+1.3333=4Nowconsidersequence2,1.Theoptimalscheduledstartingtimeofthesecondoperationwouldbed(1)=F−11(2/3)=41andthetotalexpectedcostunderthisschedulewouldbe4137(41−x)16dx+43412(x−41)16dx=1.3333+0.6666=2Theexpectedcostundersequence2,1isclearlyless.Thatis,thejobwiththelargermeanandthesmallervarianceshouldgofirst.WhathappensnowifthedistributionofY2isUniformlydistributedoverthesupport[3,9]?Nowithasamuchsmallermean,butthesamevariance.AssumethatthedistributionofY1remainsthesame.Thesameanalysisasbeforegoesthroughandsequence2,1remainsoptimal.So,inbothcasestheSmallestVariancefirst(SV)ruleisoptimalandthemeansdonotplayarole.Fromtheexampleaboveitisclearthatthevariancesofthedistributionfunctionshaveamuchstrongerinfluenceontheoptimalschedulethanthemeansofthedistributions.Inordertostudytheimpactofthevarianceontheoptimalsequenceamoretheoreticalframeworkhastobeestablished.AssumethatthetworandomvariablesY1andY2areorderedaccordingtotheso-calledvariabilityorderorconvexordering,i.e.,Y1issaidtobelessvariablethanY2if∞0g(t)dF1(t)≤∞0g(t)dF2(t)or,equivalently,E(g(Y1))≤E(g(Y2))forallconvexfunctionsg.Itfollowsfromthefactthatbothfunctionsg(t)=tandg(t)=−tareconvexthatthemeansofY1andY2areequal,i.e.,E(Y1)=E(Y2).IfY1islessvariablethanY2,thenE(Y1)=E(Y2)andVar(Y1)≤Var(Y2)(seeExercise12.2).
12.2SchedulingaSingleOperatingRoom295Considernowthetwopossiblesequences1,2and2,1.Letd(1)=adenotethevalueoftheoptimaltimetoschedulethestartofthesecondsurgeryundersequence1,2andletd(1)=bdenotethevalueoftheoptimaltimetoschedulethestartofthesecondsurgeryundersequence2,1.Letctot({1,2},a)denotethetotalexpectedcostundersequence1,2assumingd(1)=a.Now,ctot({1,2},a)≤ctot({1,2},b)=worE(max(b−Y1,0))+wsgE(max(Y1−b,0))≤worE(max(b−Y2,0))+wsgE(max(Y2−b,0))=ctot({2,1},b)So,underfairlygeneralconditionsontheshapesofthedistributionfunc-tionsF1andF2,itcanbeshownthat,assumingthescheduleralwayspromisesthesecondsurgeoninthesequencetheoperatingroomattheoptimaltime(asgivenabove),theschedulerwouldminimizethetotalcostifthesurgerywiththesmallestvariancegoesfirst.Thatis,theschedulermustfollowthentheSmallestVariancefirst(SV)rule.However,showingthatSVisoptimalinanenvironmentwithnsurgeriesisconsiderablyharder.EventhoughitappearsthatSVisarulethatwouldworkwellinpractice,itisnotthateasytoimplement,becauseitisalreadyveryhardtodeterminetheoptimaltimesatwhichthethird,fourthandallsubsequentsurgeriesshouldstart.However,theoptimalityofSVcanbeestablishedforasimplermodel.ConsidernsurgerieswithrandomdurationsY1,…,YnfromdistributionsF1,…,Fn.DistributionFjhasameanµjandavarianceσ2j.Assumethatthesedistributionshavesymmetricdensityfunctions(seeFigure12.1).Nowconsideramodelthatisslightlydifferentfromthemodelwiththetwosurg-eriesdescribedabove.Theschedulernowhastoplanaheadfornsurgeriesandhastodetermineforthesensurgeriestheanticipatedstartingtimes(thestartingtimeofthefirstsurgerybeingtime0).Theplannedstartingtimeofthe(k+1)thsurgerybasicallyconstitutesaduedateforthecompletiontimeofthekthsurgery.However,incontrasttothetwosurgerymodeldescribedabove,itisnowassumedthattheoperatingroomneverremainsidle.Ifoneofthesurgeriesiscompletedbeforetheplannedstartingtimeofthenextsurgery,thenitisassumedthatthesurgicalteamthathastoperformthenextsurgerysomehowwillbeabletoadaptitsscheduleandcanmanagetostartitsoperationassoonasthepreviousoperationhasbeencompleted.LetS(k+1)denotethestartingtimeofthesurgerythatisinthe(k+1)stpositioninthesequence.So,S(k+1)=max(d(k),C(k)),fork=1,…,n−1.However,the”scrambling”thesurgicalteamhastodowhenC(k)(<)w,thendk<(>)µ1+µ2+···+µk−1+µk.Itisintuitivethatiftheearlinesscostsarehigherthanthetardinesscosts,thenthestartingtimesshouldbescheduledearlierthanincasethetwocostsareequaltooneanotherandvice-versa.Howwouldtheoptimalsolutionofthesimplifiedmodelcomparetotheoptimalsolutionoftheoriginalmodelinwhichtheoperatingroommayendupidleattimes?Assumethesamesetofjobsintheoriginalmodelandinthemoresimplifiedmodelandassumethesameschedulebeingadoptedinthetwomodels.Assumethatthewaitingcostofasurgeonintheoriginalmodelisequaltothetardinesscostinthemoresimplifiedmodelandthatthecostofhavingtheoperatingroomidleintheoriginalmodelisequaltotheearlinesscostinthemoresimplifiedmodel.Itcanbeshownthenthattheoptimaltimesfoundforthesimplifiedmodelcanserveaslowerboundsforthe(unknown)optimaltimesfortheoriginalmodel.12.3MultipleOperatingRooms-ASetPackingFormulationMosthospitalshavemorethanoneoperatingroom.Thecollectionofoperatingroomsisoftenreferredtoastheoperatingtheatre.Schedulingtheseoperatingroomsmayberegardedaseitheraparallelmachineschedulingproblem(seeChapter5)orasabinpackingproblem(seeChapter9).Inthissectionadeterministicmodelisconsidered.Consideramodelwherethedurationsofallsurgeriesaredeterministicwithnorandomnesswhatso-ever.Assumensurgerieshavetobescheduledinmoperatingrooms.Inthismodeltimetisassumedtobediscrete.LetHdenotethenumberoftimeslotsintheplanninghorizon,e.g.,an8-hourdayisdividedinto96slotsof5min-uteseach.LetRtotdenotethesetofallpossibleresources,includingrooms,
29812PlanningandSchedulinginHealthCarestaff,andequipment.LetRCjdenotethesetofallpossiblecombinationsofresourcesthatwouldsatisfytheneedsofsurgeryj(asuitableresourcecom-binationmayconsistofaspecificsurgeon,aspecificanasthaesiologist,andaspecificoperatingroom).Asuitableresourcecombinationl∈RCjhasapreferredstartingtimetlj.LetUljdenotetherangeofthetimeslotsthestartingtimeofsurgeryjwithresourcecombinationlisallowedtodeviatefromthepreferredstartingtimetlj.Asurgeonmayhaveastrongpreferencewithregardtothestartingtimetlj;however,(s)hestillmayhavesomeflex-ibilityUlj.Forexample,Uljmaybe{−1,0,1,2,3}andtheactualstartingtimemaybetlj+u,whereu∈Ulj.TheparameterIlujrtisa0−1indicatorthattakesthevalue1ifsurgeryj,usingresourcecombinationlandstartingattimetlj+u,needsresourcerintimeslott.The0−1indicatorvariableArtdenotestheavailabilityofresourcer∈Rtotintimeslott.Letthe0−1decisionvariablexlujbe1ifsurgeryjstartsintimeslottlj+uand0otherwise.Letπlujdenotethebenefitobtainedbyhavingsurgeryjstartintimeslottlj+u.ThissurgeryschedulingproblemcannowbeformulatedasaSetPack-ingproblem.AnintegerprogrammingformulationofagenericSetPackingproblemisdescribedindetailinAppendixA(foranotherexampleofaSetPackingproblem,seethetankerschedulingproblemdescribedinChapter11).ForthesurgeryschedulingproblemtheSetPackingformulationbecomes:maximizenj=1l∈RCju∈Uljπlujxlujsubjecttol∈RCju∈Uljxluj≤1forj=1,…,nnj=1l∈RCju∈UljIlujrtxluj≤Artfort=1,…,H;r∈Rtotxluj∈{0,1}forj=1,…,n;l∈RCj;u∈UljTheobjectivefunctionmaximizesthetotalsatisfactionofthepreferencesintheschedulingofthesurgeries.Thefirstsetofconstraintsrepresentsurgeryconstraints.Anyparticularoperationmaynotbescheduledmorethanonceandonlyonecombinationofresourcesmaybeassignedtosurgeryj.Thesecondsetofconstraintsensuresthattherearenoconflictsinanytimeslottbetweenthesurgeriesasfarastheuseofresourcer∈Rtotisconcerned.Example12.3.1(SchedulingSurgeriesthroughSetPacking).Con-sideranenvironmentwiththreeoperatingrooms,threesurgeons,andthree
12.3MultipleOperatingRooms-ASetPackingFormulation299anasthaesiologists.SetRtothasthusnineelements,i.e.,r=1,…,9.Thethreeoperatingroomscorrespondtor=1,2,3,thethreesurgeonstor=4,5,6andthethreeanasthaesiologiststor=7,8,9.ThetimehorizonHconsistsof96slotsoffiveminuteseach.Thenumberofsurgeriestobescheduledisfive.surgeries12345pj4242124848Resourcecombinationl∈RCjcanbespecifiedbyatriple(x,y,z);thefirstentryofthetriplereferstotheoperatingroom,thesecondentrytothesur-geon,andthethirdentrytotheanasthaesiologist.Forexample,theresourcecombination(2,4,9)referstothesecondoperatingroomcombinedwiththefirstsurgeonandthethirdanasthaesiologist.surgeriesallowableresourcecombinations1RC1={(1,4,7),(1,4,8),(1,5,7),(3,4,7),(3,4,8)}2RC2={(1,4,7),(1,5,7),(3,4,7)}3RC3={(3,4,7),(3,4,8),(1,6,7)}4RC4={(2,4,7),(2,4,8),(3,4,7),(3,4,8)}5RC5={(2,5,8),(2,5,9),(3,5,8),(1,5,8)}WiththedataprovidedinTable12.1allthevaluesofIlujrtcanbedeter-mined.Foreachoneofthenineresources,theavailabilitydataArthavetobegivenaswell.Assumethatallthreeoperatingroomsareavailablethroughouttheday.Thefirsttwosurgeonsareavailablethroughouttheday;however,thethirdsurgeon(resource6)isnotavailableduringthefirsthouroftheday.Thefirsttwoanasthaesiologistsareavailablethroughouttheday.Thethirdanasthaesiologist(resource9)isonlyavailableduringthefirsthalfoftheday.Insteadofhavingaprofitπlujassociatedwithhavingsurgeryjstartintimeslottlj+uusingresourcecombinationl,lettherebeacostclujassociatedwithsuchdecision.TheobjectiveoftheIntegerProgramnowhastobeminimizedratherthanmaximized.Ifanysurgeryjstartsexactlyattimetlj,thenthecostcl0j=0providedresourcecombinationlisusingeitheroperatingroom1or2;ifresourcecombinationlisusingoperatingroom3,thencostcl0j=600dollars.Foreveryadditional5minutetimeintervalthatthestartingtimeofthesurgeryispostponed,thecostgoesupby10dollars.Forevery5minuteintervalthatthesurgerystartsbeforeitsmostdesiredstartingtimetljacostof20dollarsisincurred.Assumingthereexistsatleastoneschedulethatcanaccomodateallsurg-eriesthefollowingIntegerProgramcanbeformulated.minimize5j=1l∈RCju∈Uljclujxluj
30012PlanningandSchedulinginHealthCarejlRes.Comb.tljUlj11(1,4,7)0{0,1,2,3,4,5,6}12(1,4,8)48{−3,−2,−1,0,1,2,3}13(1,5,7)48{−6,−4,−2,0,2,4,6}14(3,4,7)0{0,1,2,3,4,5,6}15(3,4,8)48{−3,−2,−1,0,1,2,3}21(1,4,7)0{0,1,2,3,4,5,6}22(1,5,7)0{0,1,2,3,4,5,6}23(3,4,7)0{0,1,2,3,4,5,6}31(3,4,7)0{0,1,2,3,4,5,6}32(3,4,8)0{0,1,2,3,4,5,6}33(1,6,7)48{−6,−4,−2,0,2,4,6}41(2,4,7)48{−6,−4,−2,0,2,4,6}42(2,4,8)48{−6,−4,−2,0,2,4,6}43(3,4,7)48{−6,−4,−2,0,2,4,6}44(3,4,8)48{−6,−4,−2,0,2,4,6}51(2,5,8)0{0,1,2,3,4,5,6}52(2,5,9)0{0,1,2,3,4,5,6}53(3,5,8)0{0,1,2,3,4,5,6}54(1,5,8)0{0,1,2,3,4,5,6}Table12.1.TableforExample12.3.1.subjecttol∈RCju∈Uljxluj≤1forj=1,…,55j=1l∈RCju∈UljIlujrtxluj≤Artforr=1,…,9;t=1,…,96xluj∈{0,1}forj=1,…,5;l∈RCj;u∈UljItcanbecheckedeasilythattheobjectivefunctionhas133elements.Thefirstsetofconstraintsconsistsof5constraints.Thefirstoneoftheseconstraintscorrespondstosurgery1;thisconstraintsumsup35variables.Sincethecostofusingthethirdoperatingroomisveryhigh,theIntegerProgramwilltrytopackthe5surgeriesinoperatingrooms1and2.Eventhoughthefivesurgeriesthencannotstartattheirmostdesiredstartingtimes,thetotalcostwillbeminimized.Theoptimalsolutionassignssurgeries2,3,1tooperatingroom1(inthatorder)andsurgeries5and4tooperatingroom2(alsointhatorder).Thefirstanasthaesiologist(resource7)isused
12.4MultipleOperatingRooms-AStochasticApproach301throughoutthedayinoperatingroom1andthesecondanasthaesiologistisusedthroughoutthedayinoperatingroom2.Notethatintheformulationabove,eachoperatingroomrepresentsauniqueresource.Evenwhentwooperatingroomsareidentical,theyhavetobetreatedasdistinctentities.Onecouldcomeupwithadifferentintegerprogrammingformulationforthesameproblem.LetRtotdenotethesetofallresourcetypes.Sothenumberofelementsinthesetmaybesignificantlysmaller:ifahospitalhasthreeidenticaloperatingrooms,thenintheoriginalformulationthesetRtotwouldhaveanelementforeachoperatingroom.Inthenewformulation,thereisasingleelementfortheresourcetype”operatingroom”.However,nowtheindicatorvariableisnotanymorea0−1variable.Fortheoperatingroomresourcetyper,thevariableArtisnowthree.TheindicatorvariableIlujrtisnownotanymorea0−1variable;itmaynowassumemultiplepositiveintegervalues.Asstatedbefore,thereisasignificantamountofexperienceindealingwithsetpackingproblemsthroughbranch-and-bound.Foradiscussionofappropriatebranchingandboundingtechniques,seeSection11.2.12.4MultipleOperatingRooms-AStochasticApproachIntheprevioussectionallthesurgerytimeswereassumedtobefixedandknowninadvance,i.e.,therewasnotanyuncertainty.Inthissectionthesurgerytimesareagainassumedtoberandomvariables,ofwhichthemeansaswellasthevariancesareknowninadvance.Surgeryjhasanexpectedduration(mean)µjandavarianceσ2j.Therearemoperatingroomsavailable,whichareidentical.Theplanninghorizonisagaindiscrete,butthetimeunitisnowonedayandthelengthoftheplanninghorizonisHdays.ThereareNpdifferentspecialities,e.g,cardiology,neurology,andsoon.ThesespecialitiesaresimilartothedifferentpoolsofoperatorsinaworkforcedescribedinSection4.6.Speciality,=1,…,Np,hasondaytagivensetofoperatingroomsatitsdisposal,denotedbyMt.AnelementofMt,i.e.,anoperatingroomiassignedtospecialityondayt,isreferredtoasanOR-Dayandisdenotedbyatriplet(,i,t).EachOR-DayhasanOR-Teamwhichisateamofpersonnelthatisrequiredtoperformthesurgeries.LetJdenotethesetofallelectivesurgeriesthathavetobescheduled,letJdenotethesetofallsurgeriesofspeciality,andletJitdenotethesetofsurgeriesassignedbyspecialitytooperatingroomiondayt.Assumethatthereisaninitial(base)schedulewithanassignmentofsurgeriestoeachOR-day(,i,t).TheprocedureoutlinedbelowisdesignedtorescheduleallthesurgeriesintheoriginalscheduleinamoreefficientmannerthatwouldfreeupOperatingRoomcapacity.OneoftheobjectivesthathastobeminimizedistheexpectedovertimeinanOR-day.Sincethesurgerytimesarerandom,thereisalwaysaprobability
30212PlanningandSchedulinginHealthCarethatthelastsurgerymayresultinovertime.Sincetheexpectedovertimeshouldbeminimized,thescheduler,onpurpose,insertssomeslackintothesystem.TheplannedslackisbasedonthetotalvarianceofthedurationsofthesurgeriesscheduledforthatOR-day.ThemeanofthetotaldurationoftheplannedsurgeriesforOR-day(,i,t),i∈Mt,isµit=j∈Jitµjandthevarianceisσ2it=j∈Jitσ2j.TheplannedslacktimeδitforOR-day(,i,t)isthenδit=βj∈Jitσ2j,whereβisaparameterthataffectstheprobabilityofthesurgeriesbeingcompletedontime(i.e.,ofnoovertimeoccurring).Thevalueofβistypi-callychosenbymanagement;ahigherβresults,ontheonehand,inaloweroperatingroomutilization,but,ontheotherhand,inlessovertime.GivenasurgeryallocationJit,theovertimeonOR-day(,i,t)isdefinedasfollows:Oit=max(0,µit+δit−Tit),whereTitrepresentsthetotaltimeavailableonOR-day(,i,t),i.e,itstimecapacity.Thegoalistogenerateaschedulethatdoeswellwithregardtothefol-lowingthreeobjectives,namely(i)minimizetotalovertime,i,tOit,(ii)maximizethetotalnumberoffreeOR-days(anOR-day(,i,t)isfreeifJit=∅),(iii)maximizethetotalfreecapacity,i,tmax(0,Tit−µit−δit).Ofthesethreeobjectives,thefirstisusuallythemostimportantandthelasttheleastimportant.Ifthefirstandthirdobjectivewouldhavebeenequallyimportant,thenanappropriateobjectivewouldhavebeenthemini-mizationofthetotalplannedslacktime.FortheallocationofsurgeriestoOR-days,assumethatsurgeriesonlycanbeassignedtotheOR-daysoftheirspeciality.GivenafixedsetofOR-daysforeachspeciality,itisclearthatthegeneralproblemcanbedecomposedintoNpproblemsthatareindependentfromoneanother.Ifsuchasubproblemisconsidered,i.e.,theschedulingofthesurgeriesofoneparticularspecialitygivenafixedsetofdaysassignedtothatspeciality,
12.4MultipleOperatingRooms-AStochasticApproach303thenvariousheuristicscanbeused.Ifthemainobjectiveistominimizetheto-talovertime,thenthisproblemmayberegardedasaparallelmachineschedul-ingproblemwiththeminimizationofthemakespanasobjective.Chapter5describedtheLongestProcessingTimefirst(LPT)heuristic.Inthecurrentapplication,theLPTheuristicwouldfirstorderallthesurgeriesindecreasingorderofµj+βσj.Ittakesthenthesurgeriesfromthelistonebyone,eachtimeassigningthesurgerytakenfromthelisttotheOR-daywiththesmallestload.Thisheuristicisappropriateforminimizingthemakespanobjectiveandthereforealsosuitableforminimizingtotalovertime.If,ontheotherhand,themainobjectiveistheminimizationofthenumberofOR-daysusedbytheparticularspecialization(or,equivalently,themaximizationofthetotalnumberoffreeOR-days),thenthisproblemmayberegardedasabin-packingproblem(seeChapter9).Inthebin-packingproblemtheobjectiveistomini-mizethenumberofbinsused.Asuitableheuristic,whichthenwouldattempttominimizethenumberofOR-daysused,istheso-calledFirstFitDecreasing(FFD)heuristic.TheFFDheuristicalsoordersallthesurgeriesindecreasingorderofµj+βσj(similartotheLPTheuristic).AssumethatthecapacityofeachOR-dayisTminutes.TheFFDheuristictakesthesurgeriesfromthelistonebyone,eachtimeassigningthesurgerytakenfromthelisttotheOR-daywiththelargestsurgeryloadamongthoseOR-daysthatarestillabletoaccomodatethesurgery.BoththeLPTheuristicandtheFFDheuristicarebasedontheexpecteddurationsofthesurgeries(includingtheplannedslacktime).Thatis,themeantimeallocatedtosurgeryjisµj+βσj.However,thesetwoheuristicsdonottakeintoaccounttheinterplaybetweenthevariancesofthesurgeriesthatareassignedtothesamemachine.IfthetwoheuristicsassumethattheplannedslackforOR-day(,i,t)isβj∈Jitσj,thentheyoverestimatethetotalrequiredplannedslackforthatOR-day,sinceβj∈Jitσj>βj∈Jitσ2j=δitActually,itmaybebeneficialtomakechangesintheschedulesgeneratedbytheLPTandFFDheuristicsbytryingtocombinesurgeriesbasedontheirvariancesinordertoreducethetotalplannedslacktime.Considerthefollowingexample.Example12.4.1(AssignmentofSurgeriestoOR-days).ConsiderfoursurgeriesandtwoOR-days.Surgeries1and2bothhaveameanof100minutesandastandarddeviationof50minutes.Surgeries3and4bothhaveameanof100minutesandastandarddeviationof10minutes.Theexpecteddurationofsurgeries1and2is100+50βandofsurgeries3and4theexpecteddurationis
30412PlanningandSchedulinginHealthCare100+10β.Assumeβ=1andthatthetotaltimeavailable,i.e.,thecapacity,inanOR-dayisT=400minutes.LPTwouldassignsurgeries1and3tooneOR-dayandsurgeries2and4totheotherday.FFDwouldassignsurgeries1and2tooneOR-dayandsurgeries3and4totheother.UndertheLPTschedulethestandarddeviationofthetotaldurationofthesurgeriesineachoneofthedaysis502+102=51.Thetotalplannedslacktimeforthetwodaysistherefore102minutes.UndertheFFDschedulethetotalplannedslacktimeis502+502+102+102=84.9.So,clearly,inordertoreducethetotalplannedslacktime,itisbettertoassignthetwosurgerieswithhighvariancestoonedayandthetwosurgerieswithlowvariancestoanotherday.Ifallsurgerieswouldhavethesamemeanbutdifferentvariances,thentheexampleabovesuggeststhattheFFDheuristicwouldyieldabetterschedulethantheLPTheuristic.12.5PlanningandSchedulingRadiotherapyTreatmentsInoncologydepartments,theschedulingofradiotherapyforpatientsplaysacrucialroleinensuringthedeliveryoftherighttreatmentattherighttime.Itiswellknownthatalargeamountofradiationcanbedeliveredsafelytoatumorifitisspreadoutoverseveralweeks.Thisprocedureisusuallyreferredtoasfractionation;itsaveshealthytissuefromunnecessarydamageandgivesittimetorecover.Dosagefractionationimpliesthatthepatienthastovisitthetreatmentcenterseveraltimesaweekforagivennumberofweeks,dependentuponthetreatmentplan.Asamatteroffact,eachdoseofradiationlastsonly2-4minutes.Theschedulingofthetreatmentplansdealswiththeallocationofthepatientstotreatmentsovertime,subjecttoallthepossibleconstraints.Theobjectiveofoutpatientschedulingistofindanappointmentsystemforwhichaparticularperformancemeasureisoptimizedinaclinicalenvironment.Theschedulingofthetreatmentplanswillfollowadynamicappointmentsystem.(Appointmentschedulingoftenplaysaveryimportantroleinotherhealthcaresettingsaswell.)Thedailyuseofeachradiationdevicecanbemodeledasacollectionoftimeslotsofequalduration.Theobjectiveoftheoptimizationmodelistomaximizethenumberofpatientstobescheduled,takingintoaccountthefollowingconditions:(i)theprioritylevelassignedtothepatient(whichmaydependonthe”severity”ofthepatient’sillness),
12.5PlanningandSchedulingRadiotherapyTreatments305(ii)thenumberoftreatmentsessionsforeachpatientinaweek,(iii)thetreatmentsessionsforeachpatienthavetobedoneonconsecutivedays,(iv)eachpatienthastoundergothetreatmentforagivennumberofcon-secutiveweeks.Thefollowingnotationisused.LetJnewdenotethelistofnewpatientswhohavejustenteredthesystemandwhohavenotreceivedanytreatmentyet.SetJnewhasnpatients.Foreachnewpatientj,j=1,…,n,letwjdenotehisprioritylevelandletejdenotethenumberoftreatments(s)hemustreceiveperweek(whichthenmustbescheduledonconsecutivedays).LetJolddenotethelistofpatientsthatalreadyhavebeguntheirtreatmentplanandwhosetreatmenthastocontinueaccordingtotheplandeterminedinpreviousweeks.LetHdenotetheplanninghorizonindays,i.e.,t=1,…,H,andletmdenotethenumberoftimeslotsinaday,i.e.,i=1,…,m.(Inasense,thedifferenttimeslotsonanygivendaycouldberegardedasmidenticalmachinesinparallel.)TheH×mmatrixAkeepstrackofthetimeslotsthatalreadyhavebeenassignedtopatientsinsetJold,whobegantheirtreatmentprograminpreviousweeks.TheentryaitinthematrixAis1iftimeslotiondaytalreadyhadbeenassigned.Letthebinaryvariablezjbe1ifpatientj,j∈Jneworj=1,…,n,beginshistreatmentprograminthecurrentweekand0otherwise.Letthebinaryvariablexjitbe1ifpatientjisassignedtotimeslotiondaytandletthebinaryvariableyjitbe1ifpatientjhashisfirstappointmentintimeslotiondayt.Notethatpatientjwhostartshistreatmentplanthisweekwillreceiveejtreatmentsinhisfirstweek;onlythefirstoneoftheejtreatmentsqualifiesasthefirstappointment.Giventhesetofpatientsthatalreadyhavebeguntheirtreatmentsinthepreviousweeks,theobjectiveintheproposedmathematicalmodelistomaximizetheweightednumberofpatientsofsetJnewthatcanbescheduledinthecurrentweek.LetJnew,asubsetofJnew,denotethelistofpatientsthatcannotyetbegintheirtreatmentthisweek.SothesetJnew−Jnewdenotesthesetofpatientsthatactuallybegintheirtreatmentinthecurrentweek.Sothetotalnumberofpatientsscheduledinthecurrentweekis|Jold|+|Jnew−Jnew|.Theassumptionismadethatthetreatmentofeachpatienttakes15minutes.However,inpractice,thefirsttreatmentinaregimenrequiresmoretimethanthesubsequentones.Thisismainlyduetothefactthattheoperatorhastointroducethetreatmentparametersintotheradiationdevice.Ifyjit=1,thentimeslotiondaytisassignedtopatientjandthispatientwillundergohis/herveryfirsttreatmentinthistimeslot.Consequently,itisnecessarytoassignpatientjanauxiliarytimeslotforthesetupandthetwotimeslotshavetobeoneaftertheother.Thisconditioncanbeformalizedbyintroducingthebinaryvariablesjitasfollows:
30612PlanningandSchedulinginHealthCaresj,i−1,t=yjiti=2,…,m;j∈Jnew;t=1,…,H.Notethatyj1t=0andsjmt=0foralljandt.Thisconstraintensuresthatasetuptimeslotisassignedtoeachpatientjustbeforehis/herveryfirsttreatment.Inordertoformulateanintegerprogramthatyieldsanappropriatesched-uleanobjectivefunctionhastobeformulated.maximizenj=1mi=1Ht=1wjyjitsubjecttoait+nj=1xjit+nj=1sjit≤1fori=1,…,m;t=1,…,Hmi=1xjit≤1forj=1,…,n;t=1,…,Hmi=1Ht=1yjit≤1forj=1,…,nm−1i=1Ht=1sjit≤1forj=1,…,nmi=1Ht=1xjit=ejzjforj=1,…,nt+ej−1u=tmi=1xjiu−ejmi=1yjit≥0forj=1,…,n;t=1,…,H−ej+1sjit=yj,i+1,tforj=1,…,n;i=1,…,m−1;t=1,…,Hxjit≥yjitforj=1,…,n;i=1,…,m;t=1,…,Hxjit,yjit,sjit∈{0,1}forj=1,…,n;i=1,…,m;t=1,…,Hzj∈{0,1}forj=1,…,n.Thefirstsetofconstraintsensuresthatnotimeslotwillhavemorethanoneactivityassigned.Thesecondsetofconstraintsensuresthatpatientjcanbeassignedtoatmostonetimeslotonanygivenday.Thethirdsetofconstraintsensuresthatpatientjcanhaveatmostoncehisveryfirsttreatmentduringtheplanninghorizon.Thefourthsetofconstraints,which
12.5PlanningandSchedulingRadiotherapyTreatments307isrelatedtothethird,makessurethatatmostonceasetupwilloccurforpatientjoverthetimehorizonH.Thefifthsetofconstraintsensuresthatifpatientjbeginshistreatmentprograminthecurrentplanninghorizon,that(s)hewillundergoejtreatmentsovertheplanninghorizon.Thesixthsetofconstraintsensuresthatifatherapyisstartedinthecurrentplanninghorizon,allthetreatmentsaredonewithinejconsecutivedays.Theseventhsetofconstraintsensuresthatthefirsttimeapatientistreated,anadditionalsetuptimemustbeintheprecedingtimeslotonthesameday.NotethatintheIntegerProgrammingformulationdescribedabove,itisassumedthatthescheduleofthosepatientswhoarealreadyinthesystem(i.e.,whostartedtheirtreatmentinpreviousweeks)isbasicallycastinstone;theaitaregivendata.Onecouldimaginethattheoptimizationmayyieldbettersolutionsiftherewouldhavebeensomeflexibilityintheassignmentsofthesepatients.Example12.5.1(RadiotherapyScheduling).Considerascenariowith10timeslotseveryday.Itisassumedthatsixpatientshavealreadystartedtheirtreatmentprogramandtheirreservationsinthetimetableofthecurrentweekareshowninthetablebelow.Theejvalueofeachoneofthesepatientsis5.12345678910MondayXXXTuesdayXXXXXXWednesdayXXXXXXThursdayXXXXXXFridayXXXXXXSaturdayXXXTherearefivenewpatientsthatarewaitingtostarttheirtreatment.Theej-valuesofthesefivepatientsare5,5,5,5,4,respectively.ApplyingtheIntegerProgrammingformulationassumingthattheaitval-uesarefixedyieldsthefollowingsolutionwiththreeofthefivenewpatientsbeingaccomodated.12345678910MondayP1P1XXP2P2XP3P3TuesdayXP1XXXP2XP3XWednesdayXP1XXXP2XP3XThursdayXP1XXXP2XP3XFridayXP1XXXP2XP3XSaturdayXXX
30812PlanningandSchedulinginHealthCareHowever,iftheintegerprogrammingformulationwouldbemodified,al-lowingforsomeshiftingintheschedulesoftheoldpatients,thenabettersolutioncanbeobtainedaccomodatingfournewpatients.12345678910MondayP1P1P4P4XP2P2XP3P3TuesdayXP1P4XXXP2XP3XWednesdayXP1P4XXXP2XP3XThursdayXP1P4XXXP2XP3XFridayXP1P4XXXP2XP3XSaturdayXXXXTheproblemdescribedinthissectionisnotthemosttypicalexampleofanappointmentschedulingsystem.Appointmentschedulingsystemsarequitecommoninthehealthcareindustry,buttheonedescribedinthissectionhasaspectsthataredifferentfromtheappointmentschedulingsystemsinotherhealthcaresettings.First,thetreatmenttimesinradiotherapyarenotsubjecttoasignificantamountofrandomness.Thevarianceinthedurationsofthetreatmenttimestendstobelowerthanthevariancesofappointmentdurationsinotherhealthcaresettings.Second,apatientwhoisundergoingradiotherapytreatmentactuallyrequiresaseriesofappointmentsofwhichthetimingsdependupononeanother.Third,inthecaseofradiotherapytreatmentstherearenotanyno-shows.Inphysicians’offices,ontheotherhand,no-showsarecommonoccurrencesandaffectthewayinwhichappointmentsaremade.12.6EmergencyRoomStaffing-AConstraintProgrammingApproachThissectionfocusesontheproblemofcreatingschedulesforphysicianswhohavetobeassignedtoemergencyrooms.Emergencyroomsneedphysicianstaffingaroundtheclock.Emergencyroomsarestressfulworkplaces,socon-structing”good”schedulesforthephysiciansisveryimportant.Agoodsched-uleforaphysicianisaschedulethatsatisfiesmostofhisrequestsandprefer-enceswithregardtoahostofissues:thetotalamountofworktobeperformed,thespecifictimingoftheshifts,thesequencingoftheshifts,andsoon.Thephysicianschedulingproblemcanbedescribedasthepreparationofarosterforphysicianscoveringagivenplanningperiod,suchthateachshiftofeverydayhasaphysicianassignedtoit.Eachinstanceofthephysicianschedulingproblemissubjecttoawidevarietyofconstraintsdependentuponthecontextandthetraditionsofthehospital.Eachhospitalhasasetofrulesthathavebeendevelopedovertheyearstakingintoaccountthepreferencesofitsphysicians.Somehospitalshavespecialrulesconcerningnightshiftsandweekendshifts,whileotherhospitalshaveregulationsconcerningthespe-cialkindsofshiftsthathavetobepresentatalltimes.Thisemergencyroom
12.6EmergencyRoomStaffing-AConstraintProgrammingApproach309staffingproblemisaninterestingspecialcaseofaworkforce(orshift)schedul-ingproblem.(Workforceschedulingproblemswillbeconsideredinmoredetailinthenextchapter.)Inordertofacilitatetheformulationofaconstraintprogramforthephysi-cianschedulingproblem,itmaybeusefultoclassifythetypesofconstraintsthathavetobeincorporatedinsuchaformulation.Intheconstraintprogramdescribedbelowtherearetwotypesof(generic)constraints:theDistributionconstraintsandthePatternconstraints.TheDistributionconstraintsimposeupperand/orlowerboundsonthenumberofshiftsthatcanbeassignedeithertoasinglephysicianortoaspecificgroupofphysicianswithinagiventimeperiod.ThePatternconstraintscanbeusedtomodelpatternsofshiftsthatphysicianseitherwanttoavoidorwanttoimpose.Inordertodescribetheseconstraintsmoreformally,somenotationisneeded.Assumeatotalofmphysicians,i=1,…,m.Thesetofthesemphysicians(medicaldoctors)isdenotedbyM.LetIdenotethesetofalltheshiftsinaday.ADistributionconstraintmaystipulatethatagivensetofphysiciansMcanbeassignedtoatmostafixednumberofshiftsninasetofshiftsIduringthedaysinsetD.Twoadditionaloptionsareavailable:O1-theshiftsassignedhavetobeconsecutive;O2-noshiftisassignedduringthegivenperiodandtheconstraintmaybeignored.BothO1andO2canbeeithertrueorfalse.ThedistributionconstraintcanbewrittenasDistributionM,I,D,≤,n,O1,O2Forexample,itisveryusefultoplaceanupperboundnionthenumberofnightshiftsasinglephysician{i}canbeassignedtoinagivenweek:thesetofdayswouldbeweekdaysDwkdandthesetofshiftswouldbeallthenightshiftsInite.TheoptionsO1=trueandO2=falsemaybeusedtoensurethatthenightshiftsassignedareconsecutive.TheconstraintcanthusbeexpressedasDistribution{i},Inite,Dwkd,≤,ni,true,falseAnupperboundcouldalsohavebeenimposedonagroupofphysicians,notjustonasinglephysician.Anexampleofapatternconstraintisarulethatspecifiesthattheremustbeatleasta16hourbreakbetweenshifts;i.e.,someoneassignedtoanightshiftmustbeoffthenextdayandevening,andsomeoneassignedtoaneveningshiftmustbeoffthenextday.Inordertoexpresssuchapatternconstraintmoreformally,adetectionpattern(DP)aswellasaforbiddenpattern(FP)havetobespecified.Eachpatternisrepresentedbythesamesetofstartingdays(D)andtwoincrementvectors¯yand¯zthatcorrespondtotheDPandtheFP,respectively.TheDP-FPcombinationimpliesthatifaphysicianisassignedtoashiftinDP1ondayd+y1(d∈D)andtoashiftinDP2ondayd+y2,thenhecannotbeassignedtoanyshiftinFP1ondayd+z1,nor
31012PlanningandSchedulinginHealthCaretoanyshiftinFP2ondayd+z2.Thepatternconstraintcanbeformulatedasfollows:PatternDP,FP,D,¯y,¯zMorespecifically,therulethattheremustbeatleasta16hourbreakbetweenshiftscannowbeexpressedasfollows:Pattern[Inite],[Iday∪Ieve],D,[0],[1],whereInite,Iday,andIevearethesetsofallshiftsduringthenight,day,andevening,respectively,andwhereDisthesetofalldays.Beforeaconstraintprogramcanbeformulated,somenotationhastobeintroduced.Letxijtdenotea0−1decisionvariablethatis1whenphysicianiisassignedtoshiftjondaytandletMjtdenotethesetofallphysicianswhocanbeassignedtoshiftjondayt.Letaijtbe1ifphysicianicanbeassignedtoshiftjondaytand0otherwise.Thefollowingconstraintprogramcannowbeformulatedfortheemergencyroomstaffschedulingproblem.minimizef(xijt)subjecttoxijt≤aijtfori=1,…,m;j∈I;t∈Dxijt∈{0,1}fori=1,…,m;j∈I;t∈DDistributionConstraintsPatternConstraintsTheobjectivefunctionf(xijt)depends,ofcourse,verymuchonthepartic-ularapplication.Itisafunctionofthenumberofpreferencesthatarebeingviolated,thenumberofdissatisfiedphysicians,andthedistributionoftheworkloadovertheplanninghorizon.12.7ASurgerySchedulingandBedOccupancyLevellingSystemInanymidsizeorlargehospital,theschedulingoftheoperatingtheatre,whichoftenconsistsofmorethanadozenoperatingrooms,isamajorundertaking.Inthischapter,OperatingRoomschedulingproblemshavebeenanalyzedinisolation,i.e.,theeffectsoftheOperatingRoomschedulesonotherunitsofthehospitalhavebeenignored.Clearly,themastersurgeryschedulehasasignificantimpacton,forexample,thenumberofbedsoccupiedintheIntensiveCare(IC)unitandtheICnursingstaffaswellasonthenumberofbedsoccupiedinthemediumcareunit.
12.7ASurgerySchedulingandBedOccupancyLevellingSystem311TheVirgaJesseHospitalisamid-sizehospitalinHasselt(Belgium)withapproximately600beds,165medicaldoctors,and46surgeons.Theoperatingroomtheatreconsistsofnineoperatingrooms.EachoperatingroomisopenforelectivesurgeryfromMondaytoFridayfor8.5hours,i.e.,ithasatimeavailabilityTof510minutes.TheVirgaJesseHospitalhasdeveloped,incollaborationwiththeKatholiekeUniversiteitLeuven(KUL),asystemforgeneratingmastersurgeryscheduleswiththeobjectiveoflevellingtherateatwhichbedsarebeingoccupied.Thebedoccupancyrate,ofcourse,dependsontheLengthofStay(LOS)ofeachpatient,whichdependsonthetypeofsurgerythepatienthasundergone.Forexample,apatientrecoveringfromanappendixsurgerytypicallyleavesthehospital,say,after3days.Apatientwhohasundergoneasurgerybyspecialityleavesthehospitalτdaysafterthesurgery,implyingthat(s)heoccupiesabedforτdaysstartingatthedayofsurgery.ThegoalofthesystemistogenerateacyclicmastersurgeryschedulefortheoperatingroomsinsuchawaythattheTotalBedShortage(TBS)overtheplanninghorizon,whichisbasicallyacycle,isminimized.Recallthatacyclicscheduleisaschedulethatisrepeatedafteragiven(fixed)timeperiod,usuallyreferredtoasthecycletime(seeChapter6).Withinacycletheremaybeanumberoftimeperiodsduringwhichsurgeriescannottakeplace.Theseperiodsarereferredtoasidleperiods,whiletheotherperiodsarereferredtoasbusyperiods.Typically,cycletimesaremultiplesofweekswiththeweekendsbeingtheidleperiods.However,eventhoughnosurgeriesarescheduledduringtheweekends,bedsstillwillbeoccupiedovertheweekendbypatientswhohaveundergonesurgeriesduringtheweek.TheproblemofschedulingthedifferenttypesofsurgeriesfromthedifferentspecialitiesinacyclesuchthatthetotalbedshortageisminimizedovertheplanninghorizoncanbeformulatedasanIntegerProgramwithanonlinearobjectivefunction.Thebasicstructureofthemathematicalprogramcanbeformulatedasfollows.Letxtdenotethenumberofblocksassignedtospeciality(surgicalgroup)ondayt.Ifxt=k,thenthenumberofbedsbeingoccupiedovertheperiodt,t+1,…,t+τisktimesthenumberofpatientsoperateduponineachblockassignedtothesurgicalgroupfordayt.LetbdenotethetotalnumberofblocksrequiredbyspecialityoveracycleoflengthH.Letbtdenotethemaximumnumberofblocksavailableondayt.(Clearly,ifdaytisinanidleperiod,thenbt=0.)MinimizeTBS=Ht=1BStsubjecttoHt=1xt=bfor=1,…,Np
31212PlanningandSchedulinginHealthCareNp=1xt≤btfort=1,…,Hxt∈0,1,2,…,min(b,bt)for=1,…,Np;t=1,…,H.Theobjectivefunctionminimizesthetotalbedshortageoverthecycle.Thefirstsetofconstraintsensuresthateachspeciality(surgicalgroup)receivesitsnumberofrequiredblocks.Thesecondsetofconstraintsensuresthatthenumberofblocksassignedeachdaydoesnotexceedthetotalnumberofblocksavailable.Thelastconstraintsetensuresthatthedecisionvariablextisinteger.Notethatthebasicmodelallowsaspeciality(surgicalgroup)tohavemultipleblocksonthesameday;thisispossiblesinceasurgicalgroupmayconsistofmanysurgeons.Clearly,theMixedIntegerProgramdescribedaboveisjustabare-bonesformulation.Anactualformulationismorecomplicated.Forexample,thereisaneedforauxiliaryvariablesthatrelatethenumberofsurgeriesdonebyspecialityondaytwiththenumberofbedsoccupiedatdays,s≥t(seeExercise12.7).Actually,thesystem,asdesignedfortheVirgaJesseHospitalisbasedonaconsiderablymoresophisticatedapproachthantheonedescribedabove.Ittakesalsovarioussourcesofrandomnessintoconsideration.First,itassumesthattheLOS(τ)ofeachpatientisnotjustafixednumberofdays(deter-ministic),butratherarandomvariablethatfollowsa(discrete)multinomialdistributionwithparametersthatdependonthetypeofsurgery.Forexample,apatientrecoveringfromanappendixsurgeryleavesthehospitalafter2dayswitha20%probability,after3dayswitha50%probability,andafter4dayswitha30%.Asecondsourceofrandomnessisduetothefactthatifspecialityhasbeengivenablock,thenumberofpatientsundergoingsurgeryinthisblockisrandomaswell.SothenumberofbedsoccupiedeachdaywillbearandomvariableandtheobjectivethatthesystemmustminimizeistheTotalExpectedBedShortage(TEBS).Aswithanysystemthatisimplementedandthatisbeingusedinpractice,certainshortcomingshavebecomeapparent.First,thesystemdoesnotmakeacleardistinctionbetweenthedifferentoperatingrooms.Surgeonsfromthesamespecialitymaybeassignedindifferenttimeperiodstodifferentoperatingrooms.Actually,inmanycases,surgeonsfromthesamespecialityprefertobeassignedtothesameroom,sincesuchanassignmentwouldhavepracticalbenefits.Inanycase,theschedulercaneasilyswap,throughaveryuser-friendlyGUI,theroomassignmentswithinthesamedayinordertoassignsurgeonsfromthesamespecialityasmuchaspossibletothesameroom.Asecondshortcomingisthatintheproposedmodel,thesamescheduleisrepeatedeverycycle.Itmaybemoreefficientifthesurgeongroupswithfewhoursofoperatingtimeineachcycleaggregatethesehoursandoperateonlyonceineverytwoorthreecycles(i.e.,implyingthatdifferentspecialitiesmayenduphavingdifferentcycletimes).
12.8Discussion313Fig.12.2.UserInterfaceoftheVirgaJesseOperatingRoomSchedulingSystemFigure12.2depictsaGraphicalUserInterfaceoftheVJORSSOperatingRoomSchedulingSystemasitisimplementedattheVirgaJesseHospital.Theleftpanedepictsthesurgeryscheduleforaparticularweek.Therightpanepresentstheresultingbedoccupancylevel;eachrowofsevenredbars(oneperday)representsthebedoccupancyofaward.Notethattheunderlyingmodelinthiscaseissimilartoaflexibleflowshopwithnobuffersinbeweensuccessivestages(seeChapter5).Thisflexibleflowshopconsistsofthreestages.Thefirststageconsistsofmoperatingroomsinparallel.ThesecondstageconsistsofthenumberofbedsintheICUandthethirdstageconsistsofthenumberofbedsintheMediumCareUnit.Thereasonwhythereisnotanybufferinbetween,isthatassoonasasurgeryhasbeencompleted,thepatientmustgototheICU,andsoon.12.8DiscussionThischapterfocusesonsomeoftheverybasicplanningandschedulingprob-lemsthatoccurinhospitalsettings.However,themodelspresentedaresim-plifiedvariationsofthosemodelsthatareactuallybeingusedinpractice.Thereareseveralreasonswhytheplanningandschedulingmodelsusedinpracticearemorecomplicatedthanthosedescribedinthischapter.First,
31412PlanningandSchedulinginHealthCarethemanyplanningandschedulingprocessesinahospitalareverymuchinter-twined.Forexample,themastersurgeryscheduleimposesmanyrequirementsandconstraintsonthenursesschedule.Also,ashasbeenmadeclearintheprevioussection,themastersurgeryschedulehasasignificantimpactonthebedoccupancy.Typically,suchanenvironmentcanbemodeledeitherasaflexibleflowshoporasaflexiblejobshop.However,thisparticularenviron-menthascertainprocessingcharacteristics:forexample,wheneverajobhasbeencompletedatonestage,ithastostartitsprocessingatthenextstageverysoon(ormaybeevenimmediately)thereafter.Suchacharacteristicim-pliesthatinbetweenstagestherearebasicallynobufferspaces,wherejobsareallowedtowaitforanunlimitedamountoftime.Theoverallschedul-ingproblemis,ofcourse,alsoverycomplicatedbecausetherearemultipleobjectives.Asecondreasonwhythemodelsinpracticearemorecomplicatedisthefollowing:Thischapterhasonlyfocusedonplanned(elective)surgeriesandignoredtheunplanned(emergency)surgeries.Inpractice,mostsurgeries,say80-90%,areplanned.However,10-20%ofthesurgeriesareemergencies.Intheschedulingoftheplannedsurgeries,theschedulerhastotakeintoaccountthefactthattherewillbe,duetoemergencies,arandomdemandforOperatingRoomcapacitycominginovertime.Thereareseveralwaysfordealingwithunanticipateddemandforoperatingroomcapacity.Forexample,inamediumorlargehospitalwithmorethan10operatingrooms,theschedulermaydecidetokeepatalltimes1or2operatingroomsidleinordertobeabletodealwithemergencies.Ontheotherhand,insmallerhospitalswithjustafewoperatingrooms,theschedulerwouldliketomakesurethatatanyrandompointintime,thetimetillthenextsurgerybeingfinishedisminimized;thatis,inasensetheschedulershouldminimizethemaximumintercompletiontimebetweensurgeries.Soifatanypointintimeanemergencycomesin,thenthetimetillthenextoperatingroombecomesavailable,isminimized.Thetimethatasurgeryhasbeencompletedinanoperatingroomisapotential”break-in”point.Thesectiononplanningandschedulingradiotherapytreatmentsprovidesaniceexampleofanappointmentschedulingprocess.Thistypeofprocessisquiteimportantinavarietyofhealthcaresettings.Consider,forexample,aphysician’sofficethatreceivescallsfrompatientsrequestingappointments.Theobjectiveoftheofficeistoprovidegoodservice(ensuringthatthepatientsdonothavetowaittoolong)aswellasmaximizetheutilizationofthestaff.Theappointmentsystemsinphysiciansofficesmustcopewithaninterestingphenomenonthatmakestheprocesssignificantlymoredifficult,namelythephenomenonofno-shows.Peoplecommitthemselvestoacertaindayandtimeandthendonotshowup.Theprobabilityofano-showincreasesthefurtherinthefuturetheappointmentismade.Thesectionontheassignmentofphysicianstoemergencyroomshiftsisaninterestingexampleofastaffingproblem.Sincethisstaffingproblemissubjecttosomanyconstraintsandpreferences,aconstraintprogramming
Exercises315approachmaybethemostsuitableonehere.Thenextchapterisentirelydedicatedtoworkforcescheduling.Manyoftheproblemsettingsdescribedthererequiredifferentsolutiontechniques,e.g.,mathematicalprogramming.Planningandschedulinginhealthcaremaybenefitenormouslyfromthelargeamountsofdatathathavebeencollectedovertheyearsconcerningsurgerytimes,durationsofappointments,occurencesofno-shows,andsoon.Suchdatamayprovidesignificantinsightintotheprobabilitydistributionsoftherelevantrandomvariables.However,onehastobeverycarefulintheinterpretationofsuchdata.Overtheyears,thesedistributionsmaychangeforvariousreasons,namely,becauseofimprovementsintheavailablesurgi-caltechniques,becauseoflearningeffects,andbecauseofdifferencesinthecharacteristicsofthepoolsofpatients.Exercises12.1.Giveaclosed-formexpressionford(1)whenF1istheUniformdistribu-tionoverthesupport[a,b].12.2.ShowthatiftherandomvariablesY1andY2areconvexlyordered,thenE(Y1)=E(Y2)andVar(Y1)≤Var(Y2).12.3.ConsidertworandomvariablesX1andX2whicharebothUniformlydistributed.TherandomvariableX1hasasupport[a,b]andtherandomvariableX2hasasupport[c,d].ShowthattherandomvariableZ=X1+X2hasasymmetricdensityfunction.12.4.ConsiderExample12.3.1.Assumethatthethirdoperatingroomisnotavailableanymore.Allresourcecombinationsthatdonotrequireoperatingroom3stillremainfeasible.AssumeallotherdatainExample12.3.1remainthesame.Writeouttheintegerprogramforthisparticularexample.12.5.FormulateanalternativeintegerprogramforthemodeldescribedinSection12.3.AssumenowthatRtotrepresentsthesetofallresourcetypes.(Keepinmindthatasurgerymayrequiremorethanoneunitofthesameresourcetype.)12.6.HowdoestheIntegerProgrammingformulationinSection12.5changeiftheejtreatmentsthathavetobedonewithinaweekdonothavetobedoneonconsecutivedays(butstillhavetobedonewithinthesameweek)?12.7.FormulatetheIntegerProgramforthecasestudydescribedinSection12.6inmoredetail.AssumethattheLengthofStay(LOS)ofeachpatientisdeterministic(i.e.,notrandom).Introduceauxiliaryvariablesysthatmeasurethenumberofbedsoccupiedondaysduetosurgeriesbyspecialityondayt(t≤s).Chooseanappropriateobjectivefunction.
31612PlanningandSchedulinginHealthCare12.8.Considerahospitalwithasingleoperatingroom.TheICunithas4beds.TheoperatingroomhastobescheduledfromMondaythroughFriday;theICunitisstaffed7daysaweek.Therearetwotypesofelectivesurgeriesthathavetobescheduled.Bothtypesofsurgeriesrequirehalfadayoftheoperatingroom.However,apatientthatundergoessurgeryofthefirsttypeneedstoremaintwodaysintheICunit,whereasapatientthatundergoessurgeryofthesecondtypeneedsfourdaysofcareintheICunit.Eachweeksixsurgeriesofthefirsttypeandfoursurgeriesofthesecondtypehavetobescheduled.Whatistheoptimalschedule?CommentsandReferencesAnenormousamountofresearchhasbeendoneonplanningandschedulinginhealthcare.PierskallaandBrailer(1994)presentedanoverviewofOperationsResearchapplicationsinhealthcaredelivery.Brandeau,SainfortandPierskalla(2004)editedahandbookonOperationsResearchinhealthcare.Severalchaptersinthisbookfocusonplanningandschedulingapplicationsinhealthcare.Afairamountofresearchhasbeendoneonsurgeryscheduling,attimesalsoreferredtoasoperatingtheatrescheduling.Ithasbeenclearfromtheoutsetthatsurgeryschedulingisaformofstochasticschedulinginwhichthedurationsoftheoperationsarerandomvariablesdrawnfromknowndistributions.May,StrumandVargas(2000)didathoroughstudyindicatingthattheLognormaldistributiondoesfitoperatingroomdataquitenicely.ThefirstanalysisofasingleoperatingroomwithtwoconsecutivesurgerieswasdonebyWeiss(1990).Sincethen,thesingleoperatingroomschedulingproblemhasreceivedasignificantamountofattention;see,forexample,Denton,ViapianoandVogl(2007).ThesetpackingformulationforthemultipleoperatingroomschedulingproblemisduetoVelasquezandMelo(2006);seealsoVelasquez,Melo,andKuefer(2008).Therobustsurgeryloadingproblem(assumingrandomdurations)isduetoHans,Wullink,vanHoudenhoven,andKazemier(2008).Conforti,Guerriero,andGuido(2008)developedtheintegerprogrammingfor-mulationfortheradiotherapytreatmentproblem.AsimilarproblemhasalsobeenstudiedbyFei,Combes,Meskens,andChu(2006).Theconstraintprogrammingapproachfortheemergencyroomstaffingprob-lemwasdevelopedbyRousseau,GendreauandPesant(2002);seealsoGendreau,Ferland,Gendron,Hail,Jaumard,Lapierre,PesantandSoriano(2006).ThecasestudyrelatingthesurgeryschedulingproblemwithbedoccupancylevelsisduetoBelien(2006)andBelienandDemeulemeester(2007).AsdiscussedinSection12.6,theproblemunderlyingthiscasecanbemodeledasaflexibleflowshopwithnobufferstoragesbetweenthestages.Hsu,deMatta,andLee(2003)indeedmodeledasimilarproblemasaflexibleflowshopanddevelopedatabu-searchheuristictooptimizetheirenvironment.PhamandKlinkert(2008)modeledthesurgicalcaseschedulingasaflexiblejobshopanddevelopedaMixedIntegerProgrammingformulation.AnotherinterestingcasestudyisduetoVissers,Adan,andBekkers(2005).
Chapter13WorkforceScheduling13.1Introduction……………………………31713.2Days-OffScheduling……………………..31813.3ShiftScheduling…………………………32413.4TheCyclicStaffingProblem……………….32713.5ApplicationsandExtensionsofCyclicStaffing..32913.6CrewScheduling………………………..33113.7OperatorSchedulinginaCallCenter……….33513.8Discussion……………………………..33913.1IntroductionWorkforceallocationandpersonnelschedulingdealwiththearrangementofworkschedulesandtheassignmentofpersonneltoshiftsinordertocoverthedemandforresourcesthatvaryovertime.Theseproblemsareveryim-portantinserviceindustries,e.g.,telephoneoperators,hospitalnurses,po-licemen,transportationpersonnel(planecrews,busdrivers),andsoon.Intheseenvironmentstheoperationsareoftenprolongedandirregularandthestaffrequirementsfluctuateovertime.Theschedulesaretypicallysubjecttovariousconstraintsdictatedbyequipmentrequirements,unionrules,andsoon.Theresultingproblemstendtobecombinatoriallyhard.Inthischapterwefirstconsiderasomewhatelementarypersonnelschedul-ingproblemforwhichthereisarelativelysimplesolution.Wethendescribeanintegerprogrammingframeworkthatencompassesalargeclassofpersonnelschedulingproblems.Wesubsequentlyconsideraspecialclassoftheseinte-gerprogrammingproblems,namelythecyclicstaffingproblems.Thisclassofproblemshasmanyapplicationsinpracticeandiseasyfromacombinatorialpointofview.Wethenconsiderseveralspecialcasesandextensionsofcyclic© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_13,317
31813WorkforceSchedulingstaffing.Inthesixthsectionwediscussthecrewschedulingproblemsthatoc-curintheairlineindustry.Inthesubsequentsectionwedescribeacasethatinvolvestheschedulingofoperatorsinacallcenter.13.2Days-OffSchedulingWefirstconsiderafairlyelementarypersonnelassignmentproblem.Eachdayoftheweekanumberofemployeeshavetobepresent.Thenumbermaydifferfromdaytoday,butthesetofrequirementsremainsthesamefromweektoweek.Thereisatotalnumberofemployeesavailableandeachhastobeassignedasequenceofdays.However,theassignmentofdaystoanygivenemployeemaybedifferentfromoneweektothenext.Byaweek,wemeansevendaysthatstartwithaSundayandendwithaSaturday.Theproblemistofindtheminimumnumberofemployeestocoverasevendayaweekoperationsuchthatthefollowingconstraintsaresatisfied.(i)Thedemandperday,nj,j=1,…,7,(n1isSundayandn7isSaturday)ismet.(ii)Eachemployeeisgivenk1outofeveryk2weekendsoff.(iii)Eachemployeeworksexactly5outof7days(fromSundaytoSaturday).(iv)Eachemployeeworksnomorethan6consecutivedays.Theseconstraintscanhavecertaineffectsonthescheduleofanemployee.Forexample,ifanemployeehasoneweekendoff,thenhecannotworksixdaysstraightandhavehisnextdayoffonthefollowingSunday,becauseheviolatesthenthethirdconstraint(workingexactly5daysoutofsevendaysthatstartwithaSundayandendwithaSaturday).However,anemployeecanhaveaconsecutiveSaturday,SundayandMondayoff,aslongasheworksafterthatforatleast5daysinarow.Actually,hecouldhaveaweekendoffandtakeagainasingledayoffafter2or3daysofwork.Wenowdescribeamethodthatgeneratesanoptimalscheduleoneweekatatime,i.e.,afterthescheduleforweekihasbeenset,thescheduleforweeki+1isdetermined,andsoon.Itturnsoutthatthereexistsacyclicoptimalschedulethat,afteranumberofweeks,repeatsitself.Therearethreesimplelowerboundsontheminimumsizeoftheworkforce,W.First,thereistheweekendconstraint.Theaveragenumberofemployeesavailableeachweekendmustbesufficienttomeetthemaximumweekenddemand.Ink2weekseachemployeeisavailablefork2−k1weekends.So,assumingthat(ascloseaspossible)thesamenumberofworkersgeteachofthek2weekendsoff:(k2−k1)W≥k2max(n1,n7)andthereforeW≥k2max(n1,n7)k2−k1.
13.2Days-OffScheduling319Second,thereisthetotaldemandconstraint.Thetotalnumberofemployeedaysperweekmustbesufficienttomeetthetotalweeklydemand.Sinceeachemployeeworksfivedaysperweek,5W≥7j=1njorW≥157j=1nj.Third,wehavethemaximumdailydemandconstraintW≥max(n1,…,n7).Theminimumworkforcemustbeatleastaslargeasthelargestofthesethreelowerbounds.Inwhatfollows,wepresentanalgorithmthatyieldsaschedulethatrequiresaworkforceofasizeequaltothelargestofthesethreelowerbounds.Thealgorithmthatsolvesthisproblem,i.e.,thatfindsaschedulethatsat-isfiesallconstraintsusingthesmallestpossibleworkforce,isrelativelysimple.LetWdenotethemaximumofthethreelowerboundsandletndenotethemaximumweekenddemand,i.e.,n=max(n1,n7).Letuj=W−nj,forj=2,…,6,anduj=n−nj,forj=1and7;theujisthesurplusnumberofemployeeswithregardtodayj.Thesecondlowerboundguaranteesthat7j=1uj≥2n.Itisclearthatemployeesshouldbegivendaysoffonthosedaysthathavealargesurplusofemployees.Thealgorithmthatconstructsthescheduleusesalistofso-calledoff-daypairs.Thepairsinthislistarenumberedfrom1tonandthelistiscreatedasfollows:First,choosedayksuchthatuk=max(u1,…,u7).Second,choosedayl,(l=k),suchthatul>0;iful=0foralll=k,thenchoosel=k.Third,addpair(k,l)tothelistanddecreasebothukandulby1.Repeatthisprocedurentimes.Attheendofthelistpairsoftheform(k,k)mayappear;thesepairsarecallednondistinctpairs.Nownumbertheemployeesfrom1toW.Notethatsincethemaximumdemandduringaweekendisn,theremainingW−nemployeescanhavethatweekendoff.AssumethatthefirstdaytobescheduledfallsonaSaturday,andthefirstandseconddaysareweekend1.
32013WorkforceSchedulingAlgorithm13.2.1(Days-OffScheduling).Step1.(Scheduletheweekendsoff)AssignthefirstweekendofftothefirstW−nemployees.AssignthesecondweekendofftothesecondW−nemployees.Thisprocessiscontinuedcyclicallywithemployee1beingtreatedasthenextemployeeafteremployeeW.Step2.(Categorizationofemployeesinweek1)Inweek1eachemployeefallsintooneoffourcategories.TypeT1:weekend1off;0offdaysneededduringweek1;weekend2offTypeT2:weekend1off;1offdaysneededduringweek1;weekend2onTypeT3:weekend1on;1offdaysneededduringweek1;weekend2offTypeT4:weekend1on;2offdaysneededduringweek1;weekend2onSincethereareexactlynpeopleworkingeachweekend,|T3|+|T4|=n(becauseofweekend1);|T2|+|T4|=n(becauseofweekend2).Itfollowsthat|T2|=|T3|.PaireachemployeeofT2withoneemployeeofT3.Step3.(Assigningoff-daypairstoemployeesinweek1)Assignthenpairsfromthetopofthelist.FirsttoemployeesofT4:eachemployeeofT4getsbothdaysoff.SecondtoemployeesofT3:eachemployeeofT3getsfromhispairtheearlierdayoffandhiscompanionofT2getsfromthatsamepairthelaterdayoff.(SoeachemployeeofT3andT2getsonedayoffinweek1,asrequired.)Step4.(Assigningoff-daypairstoemployeesinweeki)Assumeaschedulehasbeencreatedforweeks1,…,i−1.AcategorizationofemployeescanbedoneforweekiinthesamewayasinStep2.Inordertoassignemployeestooff-daypairstwocaseshavetobeconsidered.Case(a):(Alloff-daypairsinthelistaredistinct)EmployeesofT4andT3areassociatedwiththesamepairsasthosetheywereassociatedwithinweeki−1.AT4employeegetsfromhispairbothdaysoff.AT3employeegetsfromthepairheisassociatedwiththeearlierdayoffandhiscompanionofT2getsfromthatpairthelaterdayoff.
13.2Days-OffScheduling321Case(b):(Notalloff-daypairsinthelistaredistinct)Weekiisscheduledinexactlythesamewayasweek1,independentofweeki−1.Seti=i+1andreturntoStep4.Thisalgorithmneedssomemotivation.First,itmaynotbeimmediatelyclearthatweneverwillbeconfrontedwiththeneedtoscheduleanondistinctpairofdaystoatypeT4worker.Inthatcaseaworkerneedstwoweekdaysoffinaweek,andwetrytogivehimthesamedaytwice.ItcanbeshownthatthenumberofT4employeesisalwayssmallerthanorequaltothenumberofdistinctpairs.Iftherearenon-distinctpairs,i.e.,pairs(k,k),intheoff-dayslist,thenweekiisindependentofweeki−1.Itcanbeshownthateverypaircontainsdaykandthemaximumworkstretchfromweeki−1toweekiis6days.Theonlytimethattheworkstretchisgreaterthan5daysiswhentherearenondistinctpairs.Inthenextexample,thereisonedistinctpair,onenondistinctpair,and|T4|=1.Example13.2.2(ApplicationofDays-OffSchedulingAlgorithm).Considertheproblemwiththefollowingdailyrequirements.dayj1234567Sun.Mon.Tues.Wed.Thurs.Fri.Sat.Requirement1033332Themaximumweekenddemandisn=2andeachpersonrequires1outof3weekendsoff,i.e.,k1=1andk2=3.SoW≥(3×2)/(3−1)=3,W≥15/5=3,W≥3.SotheminimumnumberofemployeesWis3andW−n=1.Weassignaweekendofftooneemployeeeachweek.Thisresultsinthefollowingassign-mentofweekenddaysoffforthethreeemployees.SSMTWTFSSMTWTFSSMTWTFS1XXX2XX3XXAtthispoint,thereisonesurplusemployeeonSundayandthreeonMonday.
32213WorkforceSchedulingdayj1234567uj1300000Thereare2pairsofoffdays,onedistinctandonenon-distinct.Pair1:Sunday-Monday;Pair2:Monday-Monday;Wewillusethesepairsforeachweek.Thereisonenondistinctpairinthelist.Thecategorizationoftheem-ployeesinthefirstweekresultsinthefollowingcategories:ThefirstpairisassignedtotheT4employee(oneeachweek)andthesecondpairissplitbetweentheremainingtwoemployees(typesT2andT3).Applyingthenextstepofthealgorithmyieldsthefollowingschedule.SSMTWTFSSMTWTFSSMTWTFS1XXXXXXX2XXXXXX3XXXXXXThescheduleproducesasix-dayworkstretchforoneemployeeeachweek.Thiscannotbeavoidedsincethesolutionisunique.Itcanbeshownthatifalloff-daypairsaredistinctthenthemaximumworkstretchis5days.Inthenextexamplealloff-daypairsaredistinct.Example13.2.3(ApplicationofDays-OffSchedulingAlgorithm).Considertheproblemwiththefollowingdailyrequirements.dayj1234567Sun.Mon.Tues.Wed.Thurs.Fri.Sat.Requirement3555773Themaximumweekenddemandn=3andeachpersonrequires3outof5weekendsoff,i.e.,k1=3andk2=5.SoW≥(5×3)/2=8,W≥35/5=7,W≥7.SotheminimumnumberofemployeesWis8andW−n=5.Weassignweekendsoffto5employeeseachweek.Thisresultsinthefollowingassignmentofweekenddaysofffortheeightemployees.
13.2Days-OffScheduling323SSMTWTFSSMTWTFSSMTWTFS1XXXXX2XXXXX3XXXXX4XXXXX5XXXX6XXXX7XXXX8XXXAtthispoint,thereare8peopleavailableeachweekday,sothesurplusujisdayj1234567uj0333110Thereareanumberofwaysinwhich3pairsofoffdayscanbechosen.Forexample,Pair1:Monday-Tuesday;Pair2:Tuesday-Wednesday;Pair3:Tuesday-Wednesday.Wewillusethesepairsforeachweek.Therearenonondistinctpairsonthelist.Thecategorizationoftheem-ployeesinthefirstweekresultsinthefollowing4categories:T1:1,2T2:3,4,5T3:6,7,8T4:–Employee3ispairedwith6,4with7and5with8.Thusweneedthreepairsofweekdaystogiveoff.Categorizationoftheemployeesinthesecondweekyieldsthefollowing4categories.T1:6,7T2:1,2,8T3:3,4,5T4:–Employee1ispairedwith3,2with4,and8with5.Categorizationoftheemployeesinthethirdweekresultsinthefollowing4categories.T1:3,4T2:5,6,7T3:1,2,8T4:–
32413WorkforceSchedulingEmployee1ispairedwith6,2with5,and8with7.Applyingthenextstepofthealgorithmresultsinthescheduleshownbythefollowingtable:SSMTWTFSSMTWTFSSMTWTFS1XXXXX2XXXXX3XXXXX4XXXXX5XXXXXXX6XXXX7XXXX8XXXXXXXXXXXXXXXXXXThearrowsillustratehowapairofoff-daysissharedbytwoemployees.Allpairsaredistinct,sothemaximumworkstretchis5days.Itcanbeeasilyverifiedthatthescheduleiscyclicandthecycleis8weeks.Itcanbeshownthatschedulesgeneratedbythealgorithmalwayssatisfytheconstraints.Becauseofthewaytheoff-weekendsaredistributedovertheemployees(evenly)andbecauseofthefirstlowerbound,itisassuredthateachemployeeisgivenatleastk1outofk2weekendsoff.Thateachemployeeworksexactly5daysoutoftheweek(fromSundaytoSaturday)followsimmediatelyfromthealgorithm.Thatnoemployeeworksmorethat6daysinonestretchisalittlehardertosee.Anemployeemayhaveasixdayworkstretch(butnotlonger)whentherearenon-distinctpairs(ifallpairsaredistinct,thenthelongestworkstretchis5days).Iftherearenon-distinctpairs(k,k),thendaykhastoappearinallpairs.Intheworstcase,anemployeecanbeassociatedwithpair(j,k)inweeki−1andpair(k,l)inweeki,wherej≤k≤l.Inthiscase,hewillreceiveatleastdaykoffinweeki−1aswellasinweekiwhichresultsinasixdayworkstretch.Thestretchissmallerifeitherks(t).LetE(t)=S(t)−D(t),E+(t)=max(E(t),0),E−(t)=max(−E(t),0).SoE(t)denotestheamountofsurplusattimetandE+(t)andE−(t)denotethepositiveandthenegativepartofthissurplus.LetΨ+(t)andΨ−(t)denotetherespectivepenaltycostsattimet.Theoptimizationproblemcannowbeformulatedasfollows.minx,E+,E−Ht=1Ψ−(t)E−(t)+Ψ+(t)E+(t)+Jj=1cjxj
13.8Discussion339subjecttoE+(t)−E−(t)=j:t∈[aj,bj]xj−D(t),t=1,…,HNj=1xj≤UInaddition,mostofthevariableshaveupperandlowerbounds,i.e.,xminj≤xj≤xmaxjj=1,…,N0≤E−(t)≤E−(t)maxt=1,…,H0≤E+(t)≤E+(t)maxt=1,…,Hmmini≤mi≤mmaxii=1,…Theapproachusedhereismuchfasterthanimplicittour/breakrepresen-tationapproaches,andcanhandlecomplexbreakplacementrules.TheoperatorschedulingproblemdescribedhereisamorecomplicatedversionoftheproblemdescribedinSection13.3.Thereallifeversionisevenharderbecauseofotherissuesthathavetobetakenintoaccount.Forexam-ple,acompanymayhaveoperatorswhospeakonlyEnglishandotherswhospeakbothEnglishandSpanish.So,somecallscanbehandledbyeithertypeofoperatorandothersbyonlyonetypeofoperator.IntheterminologyofChapter2,thecallsarethejobsandtheoperatorsarethemachines;theMjsetsofthejobsarenested.Theproblemsarefurthercomplicatedbyconsider-ationoflaboragreementsandpersonnelpolicies.AnexampleofsuchapolicyistheFIFOrule,thatis,ifpersonAstartshisshiftearlierthanpersonB,thenA’sfirstbreakcannotstartlaterthanB’sfirstbreak.13.8DiscussionItisinterestingtocomparethemodelsinthischapterwiththeworkforceconstrainedschedulingandtime-tablingmodelsdescribedinSection9.4.Thejobshavetobescheduledinsuchawaythatacertainobjective,e.g.,themakespan,isminimizedandatanypointintimethedemandforpeopleremainswithinthelimit.So,thereisaflexibilityintheschedulingofthejobs.Inthischapter,themodelsaresomewhatdifferent.Thereisnoflexibilityintherequirements,sincethesearegiven.However,thesizeoftheworkforceandthenumberofpeopleineachshiftarethevariables.Inpractice,personnelschedulingproblemstendtobeintertwinedwithotherfactoryschedulingproblems.Forexample,whenitisevidentthatcom-mittedshippingdatescannotbemet,extrashiftshavetobeputin,orovertimehastobescheduled.Intheliterature,thesemoreaggregateproblems(integratingmachineschedulingandpersonnelscheduling)havenotyetbeenconsidered.However,anumberofschedulingsystems,thatarecurrentlyavailableonthemarket,offermachineschedulingfeaturestogetherwithshiftschedulingfeatures.
34013WorkforceSchedulingExercises13.1.ConsiderthemodeldescribedinSection13.2.a)Explainhowalgorithm13.2.1mayresultinanoptimalschedulethatiscyclic.b)Developamethodtocomputethenumberofweeksinthecycleofanoptimalcyclicschedule.c)Arealloptimalschedulescyclic?13.2.ConsiderExample13.2.3.Notethattheschedulehasthedisadvantagethatthereisa1-dayworkstretch(e.g.,employee3worksinweek1onMonday,whileheisoffonSundayandTuesday).Considerthefollowinglistofpaireddays:Pair1:Monday-Wednesday;Pair2:Tuesday-Thursday;Pair3:Tuesday-Wednesday.Workoutthenewscheduleandobservethattheschedulehasminimum2-dayandmaximum5-dayworkstretches.However,thesurplusislessevenlydistributed.13.3.Considerthedays-offschedulingmodelofSection13.2andtheinstancewiththefollowingdailyrequirements.dayj1234567Sun.Mon.Tues.Wed.Thurs.Fri.Sat.Requirement3575573Eachpersonrequires3outofthe5weekendsoff.ApplyAlgorithm13.2.1tothisinstance.13.4.Considerthedays-offschedulingmodelofSection13.2andtheinstancewiththefollowingdailyrequirements.dayj1234567Sun.Mon.Tues.Wed.Thurs.Fri.Sat.Requirement177710113Eachpersonrequiresatleast1outofevery2weekendsoff.ApplyAlgorithm13.2.1tothisinstance.13.5.Consideraretailstorethatisopenforbusinessfrom10a.m.to8p.m.Therearefiveshiftpatterns.
Exercises341patternHoursofWorkTotalHoursCost110a.m.to6p.m.8$50.0021p.m.to9p.m.8$60.00312p.m.to6p.m.6$30.00410a.m.to1p.m.3$15.0056p.m.to8p.m.3$16.00Staffingrequirementsatthestorevariesfromhourtohour.StaffingHourRequirement10a.m.to12a.m.312a.m.to2p.m.62p.m.to4p.m.74p.m.to6p.m.76p.m.to8p.m.4a)Formulatethisproblemasanintegerprogram.b)Solvethelinearprogrammingrelaxationofthisproblem.c)Isthesolutionobtainedunderb)optimalfortheoriginalproblem?13.6.ConsidertheinstancedescribedinExercise13.4.Alltheassumptionsarestillinforcewiththeexceptionofone.Assumenowthateachpersonmustworkeveryotherweekendandmusthaveeveryotherweekendoff.(InExercise13.4itwaspossibleforapersontohaveeveryweekendoff.)a)Formulatethisinstanceasanintegerprogram.b)Solvethelinearprogramrelaxationoftheintegerprogramformulated.Roundofftheanswertothenearestintegers.c)Comparethesolutionobtainedunderb)withthesolutionobtainedinExercise13.4.13.7.Considerthe(5,7)-cyclicstaffingproblemwiththeAmatrixasdepictedinSection13.4.The¯bvectoris(4,9,8,8,8,9,4).ThefirstentrycorrespondstoaSundayandthelastentrycorrespondstoaSaturday.Thecostvector(c1,…,c7)is(6,5,6,7,7,7,7),i.e.,theleastexpensiveshiftistheonethathasbothSaturdayandSundayoff.ApplyAlgorithm13.4.1.tofindtheoptimalsolution.13.8.ConsiderApplication(i)inSection13.5.(a)Computethenumberofcolumnsinthematrix.(b)Computethenumberofcolumnsifone-dayworkstretchesarenotal-lowed(aonedayworkstretchisaworkingdaythatisprecededandfollowedbydays-off).(c)Computethenumberofcolumnsifoneandtwodayworkstretchesarenotallowed.
34213WorkforceScheduling13.9.Considerapplication(ii)inSection13.5.Assumethatthe¯cvectoris(1,1.25,1.5,1.75,2,2.25,2.5,2.75,3,1.5,1.75,2,2.25,2.5,2.75,3,3.25,3.5,2,2.25,2.5,2.75,3,3.25,3.5,3.75,4)Therequirementsvector¯bis10,10,10,10,10,10,10,10,8,8,8,8,8,8,8,8,5,5,5,5,5,5,5,5ApplyAlgorithm12.4.1tothisinstance.(Youwillneedalinearprogrammingcodetodothis,e.g.,LINDO.)13.10.ConsiderExample13.6.2.Intheseconditerationtherowprices(6.2,7.8,3,3,2)areused.However,thisisnottheonlysetoffeasiblerowprices.Considertheset(7,7,3,3,2),whichisalsofeasible.Performthenextiterationusingthissetofprices.13.11.Consideracentraldepotand5clients.Fromthedepotasingledeliveryhastobemadetoeachoneoftheclients.Theroutesthatareallowedareshowninthetablebelow.Theobjectiveistodeterminewhichtruckshouldgotoeachclientandtheroutingthatminimizesthetotaldistancetraveled.Eachcolumninthetablerepresentsapossibletruckrouteandthecjrepresentsthetotaldistanceoftheroute.Route12345678910111213cj81044214108111266510000111000000100010011000001000100011000010000101010000100101011ApplyAlgorithm13.6.1tothisinstance.CommentsandReferencesTheelementarytextbookbyNandaandBrowne(1992)coverssome(butnotall)ofthemodelsdiscussedinthischapter.Section12.2istakenfromBurnsandCarter(1985)andisbasedonthesevendaysperweek,oneshiftperdaymodel.BurnsandKoop(1987)extendthisworkandlookatthesevendaysperweek,multipleshiftsperdaymodel.Emmons(1985),EmmonsandBurns(1991)andHungandEmmons(1993)considerrelatedmodels.ThegeneralintegerprogrammingformulationconsideredinSection13.3appearsinmanyhandbooksandsurveypapers;see,forexample,thesurveypapersbyTienandKamiyama(1982)andBurgessandBusby(1992).
CommentsandReferences343ThematerialpresentedinSections13.4and13.5isprimarilybasedonthepaperbyBartholdi,OrlinandRatliff(1980).ThecrewschedulingheuristicpresentedinSection13.6comesfromthepaperbyCullen,JarvisandRatliff(1981).Manypapershavefocusedoncrewschedulingproblems;see,forexample,MarstenandShepardson(1981),Bodin,Golden,AssadandBall(1983),andStojkovic,Soumis,andDesrosiers(1998).Abranch-and-cutmethodappliedtocrewschedulingisdescribedinHoffmanandPadberg(1993).TheairlinecrewrecoveryproblemisdescribedinLettovsky,Johnson,andNemhauser(2000).Adescriptionoftheoperatorschedulingsystemdesignedforalong-distancetelephonecompanywaspresentedatanationalmeetingoftheINFORMSsocietyinWashington,D.C.,seeGawande(1996).
PartIVSystemsDevelopmentandImplementation14SystemsDesignandImplementation……………….34715AdvancedConceptsinSystemsDesign……………..37316WhatLiesAhead?……………………………..399
Chapter14SystemsDesignandImplementation14.1Introduction……………………………34714.2SystemsArchitecture…………………….34814.3Databases,ObjectBases,andKnowledge-Bases.35014.4ModulesforGeneratingPlansandSchedules…35514.5UserInterfacesandInteractiveOptimization…35814.6GenericSystemsvs.Application-SpecificSystems……………………………….36414.7ImplementationandMaintenanceIssues…….36714.1IntroductionAnalyzingaplanningorschedulingproblemanddevelopingaprocedurefordealingwithitonaregularbasisis,intherealworld,onlypartofthestory.Theprocedurehastobeembeddedinasystemthatenablesthedecision-makertoactuallyuseit.Thesystemhastobeintegratedintotheinformationsystemoftheorganization,whichcanbeaformidabletask.Thischapterdealswithsystemdesignandimplementationissues.Thenextsectionpresentsanoverviewoftheinfrastructureoftheinfor-mationsystemsandthearchitectureofthedecisionsupportsystemsinanenterprise.Wefocusonplanningandschedulingsystemsinparticular.Thethirdsectioncoversdatabase,objectbase,andknowledge-baseissues.Thefourthsectiondescribesthemodulesthatgeneratetheplansandschedules,whilethefifthdiscussesuserinterfaceissuesandinteractiveoptimization.Thesixthsectiondescribestheadvantagesanddisadvantagesofgenericsystemsandapplication-specificsystems,whilethelastsectiondiscussesimplementa-tionandmaintenanceissues.Itis,ofcourse,impossibletocovereverythingconcerningthetopicsmen-tionedabove.Manybookshavebeenwrittenoneachofthesetopics.This© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_14,347
34814SystemsDesignandImplementationchapterfocusesonlyonsomeofthemoreimportantissuesconcerningthedesign,developmentandimplementationofplanningandschedulingsystems.14.2SystemsArchitectureTovisualizetheinformationflowthroughanorganizationoneoftenusesareferencemodel,whichdepictsalltherelevantinformationflowsinanen-terprise.AnexampleofasimplifiedreferencemodelispresentedinFigure14.1FacilityInformation managementManufacturing engineeringProduction managementShopTask managementResource allocationCellTask analysisBatch managementSchedulingDispatchingMonitoringWorkstationSetupEquipment taskingTakedownEquipmentMachiningMeasurementHandlingTransportStorageRobotcartMillingmachinePartbufferPartbufferInspectionmachinesConveyorRobotRobotRobotMillingWorkstationInspectionworkstationMaterial-handlingworkstationFacilityMachiningshopAssemblyshopVirtualcell #NVirtualcell #1Fig.14.1.SimplifiedVersionofReferenceModelNowadays,therearecompaniesthatspecializeinthedesignanddevel-opmentofsoftwarethatcanserveasabackboneforanenterprise-widein-formationsystem.Eachdecisionsupportsystem,oneverylevelintheenter-prise,canthenbelinkedtosuchabackbone.Suchaframeworkfacilitatestheconnectivityofallthemodulesinanenterprise-wideinformationsystem.AcompanythatisveryactiveinthistypeofsoftwaredevelopmentisSAP,whichisheadquarteredinWalldorf(Germany).AsdescribedinSection1.2,aplanningorschedulingsystemusuallyhastointeractwithanumberofdifferentsystemsinanorganization.Itmayreceiveinformationfromahigherlevelsystemthatprovidesguidelinesfortheactions
14.2SystemsArchitecture349OrdermasterfileShop floordata collectionsystemPerformanceevaluationScheduleeditorGraphics interfaceDatabase managementAutomatic schedulegeneratorUserFig.14.2.ConfigurationofaSchedulingSystemtobetakenwithregardtolongtermplans,mediumtermplans,shorttermschedules,workforceallocations,preventivemaintenance,andsoon.ItmayinteractwithaMaterialRequirementsPlanning(MRP)systeminordertodetermineproperreleasedatesforthejobs.Asystemmayalsointeractwithashopfloorcontrolsystemthatprovidesup-to-dateinformationconcerningavailabilityofmachines,progressofjobs,andsoon(seeFigures1.1and14.2).Aplanningorschedulingsystemtypicallyconsistsofanumberofdifferentmodules.Thevarioustypesofmodulescanbecategorizedasfollows:(i)database,objectbase,andknowledge-basemodules,(ii)modulesthatgeneratetheplans,schedulesortimetablesand(iii)userinterfacemodules.Thesemodulesplayacrucialroleinthefunctionalityofthesystem.Significanteffortisrequiredtomakeafactory’sdatabasesuitableforinputtoaplanningorschedulingsystem.Makingadatabaseaccurate,consistent,andcompleteofteninvolvesthedesignofaseriesofteststhedatamustpassbeforeitcanbeused.Adatabasemanagementmodulemayalsobeabletomanipulatethedata,performvariousformsofstatisticalanalysisandallowthedecision-maker,throughsomeuserinterface,toseethedataingraphicalform.Some
35014SystemsDesignandImplementationsystemshaveaknowledge-basethatisspecificallydesignedforplanningorschedulingpurposes.Aknowledge-basemaycontain,inoneformatoranother,alistofrulesthathavetobefollowedinspecificsituationsandmaybealsoalistofobjectsrepresentingorders,jobs,andresources.Aknowledge-basemayattimesalsotaketheformofaconstraintstorewhichcontainstheconstraintsthathavetobesatisfiedbytheplansorschedules.However,fewsystemshaveaseparateknowledge-base;aknowledge-baseisusuallyembeddedinthemodulethatgeneratestheplansorschedules.Themodulethatgeneratestheplansorschedulestypicallycontainsasuit-ablemodelwithobjectivefunctions,constraintsandrules,aswellasheuristicsandalgorithms.Userinterfacemodulesareimportant,especiallywithregardtotheimple-mentationprocess.Withoutanexcellentuserinterfacethereisagoodchancethat,regardlessofitscapabilities,thesystemwillbetoounwieldytouse.UserinterfacesoftentaketheformofanelectronicGanttchartwithtablesandgraphsthatenableausertoeditaplanorschedulegeneratedbythesystemandtakelastminuteinformationintoaccount(seeFigure14.2).Afterauserhasadjustedtheplanorschedulemanually,heisusuallyabletofollowtheimpactofhischangesontheperformancemeasures,comparedifferentsolutions,andperform”whatif”analyses.14.3Databases,ObjectBases,andKnowledge-BasesThedatabasemanagementsubsystemmaybeeitheracustom-madeoracom-mercialsystem.Anumberofthecommercialdatabasesystemsavailableonthemarkethaveproventobeusefulforplanningandschedulingsystems.TheseareusuallyrelationaldatabasesincorporatingStucturedQueryLan-guage(SQL).ExamplesofsuchdatabasemanagementsystemsareOracleandSybase.Whetheradatabasemanagementsubsystemiscustommadeorcommer-cial,itneedsanumberofbasicfunctions,whichincludemultipleediting,sort-ingandsearchingroutines.Beforegeneratingaplanoraschedule,adecision-makermaywanttoseecertainsegmentsofanordermasterfileandcollectsomestatisticswithregardtotheordersandtherelatedjobs.Actually,attimes,hemaynotwanttofeedallthejobsoractivitiesintotheplanningandschedulingroutines,butratherasubset.Withinthedatabaseadistinctioncanbemadebetweenstaticanddynamicdata.Staticdataincludejoboractivitydataandmachineorresourcedatathatdonotdependontheplanorschedule.Somejobdatamaybespecifiedinthecustomer’sorderform,suchastheorderedproductquantity(whichisproportionaltotheprocessingtimesofalltheoperationsassociatedwiththejob),thecommittedshippingdate(theduedate),thetimeatwhichallnecessarymaterialisavailable(thereleasedate)andpossiblysomeprocessing(precedence)constraints.Thepriorities(weights)ofthejobsarealsostatic
14.3Databases,ObjectBases,andKnowledge-Bases351dataastheydonotdependontheplanorschedule.Havingdifferentweightsfordifferentjobsisusuallyanecessity,butdeterminingtheirvaluesisnotthateasy.Inpractice,itisseldomnecessarytohavemorethanthreepriorityclasses;theweightsarethen,forexample,1,2and4.Thethreepriorityclassesaresometimesdescribedas“hot”,“veryhot”and“hottest”dependentuponthelevelofmanagerpushingthejob.Theseweightsactuallyhavetobeenteredmanuallybythedecision-makerintotheinformationsystemdatabase.Todeterminetheprioritylevel,thepersonwhoenterstheweightmayusehisownjudgement,ormayuseaformulathattakesintoaccountcertaindatafromtheinformationsystem(forinstance,totalannualsalestothecustomerorsomeothermeasureofcustomercriticality).Theweightofajobmayalsochangefromonedaytoanother;ajobthatisnoturgenttoday,maybeurgenttomorrow.Thedecision-makermayhavetogointointothefileandchangetheweightofthejobbeforegeneratinganewplanorschedule.Staticmachinedataincludemachinespeeds,scheduledmaintenancetimes,andsoon.Theremayalsobestaticdatathatarebothjobandmachinedependent,e.g.,thesetuptimebetweenjobsjandkassumingthesetuptakesplaceonmachinei.Thedynamicdataconsistsofallthedatathataredependentupontheplanorschedule:thestartingtimesandcompletiontimesofthejobs,theidletimesofthemachines,thetimesthatamachineisundergoingsetups,thesequencesinwhichthejobsareprocessedonthemachines,thenumberofjobsthatarelate,thetardinessesofthelatejobs,andsoon.Thefollowingexampleillustratessomeofthesenotions.Example14.3.1(OrderMasterFileinaPaperMill).ConsiderthepapermilldescribedinExample1.1.4.Theordermasterfileoftherollsonordermaycontainthefollowingdata:ORDERCUSTOMERCMTWDTBWGRFNSHQTYDDTPRDTPUR01410UZSOYCO16.05.029.055.005/2505/24PUR01411UZSOYCO16.04.029.020.005/2505/25PUR01412UZSOYCO16.04.029.035.006/01TAM01712CYLEELTDPR14.03.021.07.505/2805/23TAM01713CYLEELTD14.03.021.045.005/2805/23TAM01714CYLEELTD16.03.021.050.006/07EOR01310LENSTRANVHLD16.03.023.027.506/15Eachorderischaracterizedbyan8digitalphanumericordernumber.Acus-tomermayplaceanumberofdifferentorders,eachonerepresentingaspecifictypeofrollthatischaracterizedbyasetofparameters.Therearethreepa-rameterswithregardtothetypeofpaper,namelythebasisweight(BW),thegrade(GR)andthefinish(FNS),andoneparameterwithregardtothesize,namelythewidth(WDT)oftheroll.Thequantity(QTY)orderedisspecifiedinpoundsortons.Giventhequantityandthewidththeuserofthesystemcancomputethenumberandthediameteroftherolls.Heactuallyhassomefreedomhere:hecaneithergoforasmallernumberofrollswitha
35214SystemsDesignandImplementationlargerdiameteroralargernumberofrollswithasmallerdiameter.Theupperandlowerboundsonthediametersoftherollsaredeterminedbyequipmentlimitations.ThemonthanddayofthecommittedshippingdatearespecifiedintheDDTcolumn.ThemonthanddayofthecompletiondateoftheorderarespecifiedinthePRDTcolumn;thedaysspecifiedinthiscolumncanbeeitheractualcompletiondatesorscheduledcompletiondates.Thecomments(CMT)columnisoftenempty.Ifacustomercallsandputsanorderonhold,thenHLDisenteredinthiscolumnandthedecision-makerknowsthatthisordershouldnotyetbescheduled.Ifanorderhasahighpriority,thenPRisenteredinthiscolumn.Theweightswillbeafunctionoftheseentries,i.e.,ajobonholdhasalowweight,apriorityjobhasahighweightandthedefaultvaluemaycorrespondtoanaverageweight.Setuptimesmayberegardedeitherasstaticorasdynamicdata,depend-ingonhowtheyaregenerated.Setuptimesmaybestoredinatablesothatwheneveraparticularsetuptimeneedstobeknown,thenecessarytableen-tryisretrieved.However,thismethodisnotveryefficientifthesetisverylargeandifrelativelyfewtablelook-upsarerequired.Thesizeofthematrixisn2andallentriesofthematrixhavetobecomputedbeforehand,whichmayrequireconsiderableCPUtimeaswellasmemory.Analternativewaytocomputeandretrievesetuptimes,thatismoreefficientintermsofstoragespaceandthatcanbemoreefficientintermsofcomputationtime,isthefollowing.Anumberofparameters,saya(1)ij,…,a(l)ij,maybeassociatedwithjobjandmachinei.Theseparametersarestaticdataandmayberegardedasgivenmachinesettingsnecessarytoprocessjobjonmachinei.Thesetuptimebetweenjobsjandkonmachinei,sijk,isaknownfunctionofthe2lparametersa(1)ij,…,a(l)ij,a(1)ik,…,a(l)ik.Thesetuptimeusuallyisafunctionofthedifferencesinmachinesettingsforjobsjandkandisdeterminedbyproductionstandards.Example14.3.2(SequenceDependentSetupTimes).Assumethatinordertostartajobonmachinei,threemachinesettingshavetobefixed(forexample,thecolor,thegradeandthebasisweightofthepaperinthepapermillofExample1.1.4).Sothetotalsetuptimesijkdependsonthetimeittakestoperformthesethreechangeoversandisafunctionofsixparameters,i.e.,a(1)ij,a(2)ij,a(3)ij,a(1)ik,a(2)ik,a(3)ik.Ifthethreechangeovershavetobedonesequentially,thenthetotalsetuptimeis
14.3Databases,ObjectBases,andKnowledge-Bases353sijk=h(1)i(a(1)ij,a(1)ik)+h(2)i(a(2)ij,a(2)ik)+h(3)i(a(3)ij,a(3)ik).Ifthethreechangeoverscanbedoneinparallel,thenthetotalsetuptimeissijk=maxh(1)i(a(1)ij,a(1)ik),h(2)i(a(2)ij,a(2)ik),h(3)i(a(3)ij,a(3)ik).Ofcourse,theremaybesituationswheresomeofthechangeoverscanbedoneinparallelwhileothershavetobedoneinseries.Ifthesetuptimesarecomputedthisway,theymaybeconsidereddynamicdata.Thetotaltimeneededforcomputingsetuptimesinthismannerdependsonthetypeofalgorithm.Ifadispatchingruleisusedtodetermineagoodschedule,thismethod,basedon(static)jobparameters,isusuallymoreeffi-cientthanthetablelook-upmethodmentionedearlier.However,ifsomekindoflocalsearchroutineisused,thetablelook-upmethodwillbecomemoretimeefficient.ThedecisiononwhichmethodtousedependsontherelativeimportanceofmemoryversusCPUtime.Thecalendarfunctionisoftenalsopartofthedatabasesystem.Itcontainsinformationwithregardtoholidays,numberofshiftsavailable,scheduledma-chinemaintenance,andsoon.Calendardataaresometimesstatic,e.g.,fixedholidays,andsometimesdynamic,e.g.,preventivemaintenanceshutdowns.Someofthemoremodernplanningandschedulingsystemsmayrelyonanobjectbaseinadditionto(orinsteadof)adatabase.Oneofthemainfunctionsoftheobjectbaseistostorethedefinitionsofallobjecttypes,i.e.,itfunctionsasanobjectlibraryandinstantiatestheobjectswhenneeded.Inaconventionalrelationaldatabase,adatatypecanbedefinedasaschemaofdata;forexample,adatatype“job”canbedefinedasinFigure14.3.aandaninstancecanbeasinFigure14.3.b.Objecttypesandcorrespondinginstancescanbedefinedinthesameway.Forexample,anobjecttype“job”canbedefinedandcorrespondingjobinstancescanbecreated.Allthejobinstanceshavethenthesametypeofattributes.Therearetwocrucialrelationshipsbetweenobjecttypes,namely,the“is-a”relationshipandthe“has-a”relationship.Anis-arelationshipindicatesageneralizationandthetwoobjecttypeshavesimilarcharacteristics.Thetwoobjecttypesaresometimesreferredtoasasubtypeandasupertype.Forexample,a“machine”objecttypemaybeaspecialcaseofa“resource”objecttypeanda“tool”objecttypemaybeanotherspecialcaseofaresourceobjecttype.A“has-a”relationshipisanaggregationrelationship;oneobjecttypecontainsanumberofotherobjecttypes.A“workcenter”objectmaybecomposedofseveralmachineobjectsanda“plant”objectmaycompriseanumberofworkcenterobjects.A“routingtable”objectmayconsistofjobobjectsaswellasofmachineobjects.Objecttypesrelatedbyis-aorhas-arelationshipshavesimilarcharac-teristicswithregardtotheirattributes.Inotherwords,alltheattributesofasupertypeobjectareusedbythecorrespondingsubtypes.Forexample,amachineobjecthasalltheattributesofaresourceobjectanditmayalsohave
35414SystemsDesignandImplementationIDNameTypeQuantityPriorityReadyDue2IBM4160210200NameTypeQuantityPriorityReadyDuecreatecopydeletedisplayJob IDName = IBMType = 4Quantity = 160Priority = 2Ready = 10Due = 200createcopydeletedisplayID = 2Fig.14.3.JobDataType,JobInstance,andJobObjectTypesomeadditionalattributes.Thisisoftenreferredtoasinheritance.Ahierar-chicalstructurethatcomprisesallobjecttypescanbeconstructed.ObjectscanberetrievedthroughcommandsthataresimilartoSQLcommandsinrelationaldatabases.Whilevirtuallyeveryplanningandschedulingsystemreliesonadatabaseoranobjectbase,notmanysystemshaveamodulethatservesspecificallyasaknowledge-base.However,knowledge-basesorconstraintstoresmaybecomemoreandmoreimportantinthefuture.Theoverallarchitectureofasystem,inparticularthemodulethatgen-eratestheplansorschedules,influencesthedesignofaknowledge-base.Themostimportantaspectofaknowledge-baseistheknowledgerepresentation.Oneformofknowledgerepresentationisthroughrules.Thereareseveralfor-matsforstatingrules.AcommonformatisthroughanIF-THENstatement.Thatis,IFagivenconditionholds,THENaspecificactionhastobetaken.Example14.3.3(IF-THENRuleforParallelMachines).Considerasettingwithmachinesinparallel.Themachineshavedifferentspeeds;somearefastandothersareslow.Thejobsaresubjecttosequencedependentsetuptimesthatareindependentofthemachines,i.e.,setupstakethesameamountoftimeonafastmachineastheytakeonaslowmachine.
14.4ModulesforGeneratingPlansandSchedules355Becauseofthesetuptimesitisadvisabletoassignthelongerjobstothefastermachineswhilekeepingtheshorterjobsontheslowermachines.Onecouldestablishathresholdvalueandassignthelongerjobstoafastmachine.IFajob’sprocessingtimeislongerthanagivenvalue,THENthejobmaybeassignedtoafastmachine.ItiseasytocodesucharuleinaprogramminglanguagesuchasC++.Anotherformatforstatingrulesisthroughpredicatelogicthatisbasedonpropositionalcalculus.AnappropriateprogramminglanguagefordealingwithrulesinthisformatisProlog.Example14.3.4(LogicRuleforParallelMachines).Considertheruleinthepreviousexample.APrologversionofthisrulemaybe:MACHINEOK(M,L):−longjob(L),fastmachines(F),member(M,F).TheMreferstoaspecificmachine,theLtoalongjob,andtheFtoalistofallfastmachines.The“:−”maybereadas“if”andthe“,”maybereadas“and”.Atranslationoftherulewouldbe:machineMissuitableforjobLifLisalongjob,ifsetFisthesetoffastmachinesandifmachineMisamemberofF.Asstatedbefore,thedesignofthemodulethatgeneratestheplansortheschedulesaffectsthedesignofaknowledge-base.Thisisdiscussedinmoredetailinthenextsection.14.4ModulesforGeneratingPlansandSchedulesCurrentplanningandschedulingtechniquesareanamalgamationofseveralschoolsofthoughtthathavebeenconverginginrecentyears.Oneschoolofthought,predominantlyfollowedbyindustrialengineersandoperationsresearchers,isattimesreferredtoasthealgorithmicortheoptimizationapproach.Asecondschoolofthought,thatisoftenfollowedbycomputerscientistsandartificialintelligenceexperts,includetheknowledge-basedandtheconstraintprogrammingapproaches.Recently,thetwoschoolsofthoughthavestartedtoconvergeandthedifferenceshavebecomeblurred.Somehy-bridsystemscombineaknowledgebasewithfairlysophisticatedheuristics;othersystemshaveonesegmentoftheproceduredesignedaccordingtotheoptimizationapproachandanothersegmentaccordingtotheconstraintpro-grammingapproach.Example14.4.1(ArchitectureofaPlanningandSchedulingSys-temforaWaferFab).Ahybridplanningandschedulingsystemhasbeendesignedforaparticularsemiconductorwaferfabricationunitasfol-lows.Thesystemconsistsoftwolevels.Thehigherleveloperatesaccording
35614SystemsDesignandImplementationtoaknowledge-basedapproach.Thelowerlevelisbasedonanoptimizationapproach;itconsistsofalibraryofalgorithms.Thehigherlevelperformsthefirstphaseoftheplanningandschedulingprocess.Atthislevel,thecurrentstatusoftheenvironmentisanalyzed.Thisanalysistakesintoconsiderationduedatetightness,bottlenecks,andsoon.Therulesembeddedinthishigherleveldetermineforeachsituationthetypeofalgorithmthathastobeusedatthelowerlevel.Thealgorithmicapproachusuallyrequiresamathematicalformulationoftheproblemthatincludesobjectivesandconstraints.Thealgorithmcouldbebasedonanyoneoracombinationoftechniques.The”quality”ofthesolutionisbasedonthevaluesoftheobjectivesandperformancecriteriaofthegivenschedule.Thisformofsolutionmethodmayconsistofthreephases.Inthefirstphase,acertainamountofpreprocessingisdone,wheretheprobleminstanceisanalyzedandanumberofstatisticsarecompiled,e.g.,theaverageprocessingtime,themaximumprocessingtime,theduedatetightness.Thesecondphaseconsistsoftheactualalgorithmsandheuristics,whosestructuremaydependonthestatisticscompiledinthefirstphase(forexample,inthewaythelook-aheadparameterKintheATCrulemaydependontheduedatetightnessandduedaterangefactors).Thethirdphasemaycontainapostprocessor.Thesolutionthatcomesoutofthesecondphaseisfedintoaproceduresuchassimulatedannealingortabu-search,inordertoseeifimprovementscanbeobtained.ThistypeofsolutionmethodisusuallycodedinaprocedurallanguagesuchasFortran,PascalorC.Theknowledge-basedandconstraintprogrammingapproachesareinvari-ousrespectsdifferentfromthealgorithmicapproach.Theseapproachesareof-tenmoreconcernedwithunderlyingproblemstructuresthatcannoteasilybedescribedinananalyticalformat.Inordertoincorporatethedecisionmaker’sknowledgeintothesystem,objects,rulesorconstraintsareused.Theseap-proachesareoftenusedwhenitisonlynecessarytofindafeasiblesolutiongiventhemanyrulesorconstraints;however,assomeplansorschedulesareranked”morepreferable”thanothers,heuristicsmaybeusedtoobtaina”morepreferred”planorschedule.Throughaso-calledinferenceengine,suchanapproachtriestofindsolutionsthatdonotviolatetheprescribedrulesandsatisfythestatedconstraintsasmuchaspossible.ThelogicbehindtheschedulegenerationprocessisoftenacombinationofinferencingtechniquesandsearchtechniquesasdescribedinAppendixesCandD.Theinferencingtechniquesareusuallyso-calledforwardchainingandbackwardchainingal-gorithms.Aforwardchainingalgorithmisknowledgedriven.Itfirstanalyzesthedataandtherulesand,throughinferencingtechniques,attemptstocon-structafeasiblesolution.Abackwardchainingalgorithmisresultoriented.Itstartsoutwithapromisingsolutionandattemptstoverifywhetheritisfeasible.Wheneverasatisfactorysolutiondoesnotappeartoexistorwhenthesystem’suserthinksthatitistoodifficulttofind,theusermaywanttoreformulatetheproblembyrelaxingsomeoftheconstraints.Therelaxation
14.4ModulesforGeneratingPlansandSchedules357ofconstraintsmaybedoneeitherautomatically(bythesystemitself)orbytheuser.Becauseofthisaspect,theknowledge-basedapproachhasalsobeenreferredtoasthereformulativeapproach.Theprogrammingstyleusedforthedevelopmentofknowledge-basedsys-temsisdifferentfromtheonesusedforsystemsbasedonalgorithmicap-proaches.Theprogrammingstylemaydependontheformoftheknowledgerepresentation.IftheknowledgeisrepresentedintheformofIF-THENrules,thenthesystemcanbecodedusinganexpertsystemshell.Theexpertsystemshellcontainsaninferenceenginethatiscapableofdoingforwardchainingorbackwardchainingoftherulesinordertoobtainafeasiblesolution.Thisapproachmayhavedifficultieswithconflictresolutionanduncertainty.Iftheknowledgeisrepresentedintheformoflogicrules(seeExample13.3.4),thenPrologmaybesuitable.Iftheknowledgeisrepresentedintheformofframes,thenalanguagewithobjectorientedextensionsisrequired,e.g.,C++.Theselanguagesemphasizeuser-definedobjectsthatfacilitateamodularprogram-mingstyle.ExamplesofsystemsthataredesignedaccordingtoaconstraintprogrammingapproacharedescribedinChapter10andinAppendixD.Algorithmicapproachesaswellasknowledge-basedapproacheshavetheiradvantagesanddisadvantages.Analgorithmicapproachhasanedgeif(i)theproblemallowsforacrispandprecisemathematicalformulation,(ii)thenumberofjobsinvolvedislarge,(iii)theamountofrandomnessintheenvironmentisminimal,(iv)someformofoptimizationhastobedonefrequentlyandinrealtime,(v)thegeneralrulesareconsistentlybeingfollowedwithouttoomanyexceptions.Adisadvantageofthealgorithmicapproachisthatiftheoperatingenvi-ronmentchanges(forexample,certainpreferencesonassignmentsofjobstomachines),thereprogrammingeffortmaybesubstantial.Theknowledge-basedandconstraintprogrammingapproachesmayhaveanedgeifonlyfeasibleplansorschedulesareneeded.Somesystemdevelopersbelievethatchangesintheenvironmentorintherulesorconstraintscanbemoreeasilyincorporatedinasystemthatisbasedonsuchanapproachthaninasystemthatisbasedonthealgorithmicapproach.Others,however,believethattheeffortrequiredtomodifyanysystemismainlyafunctionofhowwellthecodeisorganizedandwritten;theeffortrequiredtomodifydoesnotdependthatmuchontheapproachused.Adisadvantageoftheknowledge-basedandconstraintprogrammingap-proachesisthatobtainingreasonableplansorschedulesmayrequireincertainsettingssubstantiallymorecomputertimethananalgorithmicapproach.Inpracticecertainplanningorschedulingsystemshavetooperateinnear-realtime(itisverycommonthatplansorschedulesmustbegeneratedwithinminutes).Theamountofavailablecomputertimeisanimportantfactorintheselec-tionofaschedulegenerationtechnique.Thetimeallowedtogenerateaplan
35814SystemsDesignandImplementationoraschedulevariesfromapplicationtoapplication.Manyapplicationsrequirerealtimeperformance:aplanorschedulehastobegeneratedinsecondsorminutesontheavailablecomputer.Thismaybethecaseifreschedulingisrequiredmanytimesadaybecauseofscheduledeviations.Itwouldalsobetrueiftheplanningorschedulingenginerunsiteratively,requiringhumaninteractionbetweeniterations(perhapsforadjustmentsofworkcentercapac-ities).However,someapplicationsdoallowovernightnumbercrunching.Forexample,ausermaystartaprogramattheendofthedayandexpectanoutputbythetimeheorshearrivesatworkthenextday.Afewapplicationsrequireextensivenumbercrunching.When,intheairlineindustry,quarterlyflightscheduleshavetobedetermined,theinvestmentsatstakearesuchthataweekofnumbercrunchingonamainframeisfullyjustified.Asstatedbefore,thetwoschoolsofthoughthavebeenconvergingandmanyplanningandschedulingsystemsthatarecurrentlybeingdesignedhaveelementsofboth.OnelanguageofchoiceisC++asitisaneasylanguageforcodingalgorithmicproceduresanditalsohasobject-orientedextensions.14.5UserInterfacesandInteractiveOptimizationTheuserinterfacesareveryimportantpartsofthesystem.Theinterfacesusuallydeterminewhetherthesystemisgoingtobeusedornot.Mostuserinterfaces,whetherthesystemisbasedonaworkstationorPC,makeextensiveuseofwindowmechanisms.Theuseroftenwantstoseeseveraldifferentsetsofinformationatthesametime.Thisisthecasenotonlyforthestaticdatathatisstoredinthedatabase,butalsoforthedynamicdatathatdependontheplanorschedule.Someuserinterfacesallowforextensiveuserinteraction.Adecision-makermaymodifythecurrentstatusorthecurrentinformation.Otheruserinter-facesmaynotallowanymodifications.Forexample,aninterfacethatdisplaysthevaluesofalltherelevantperformancemeasureswouldnotallowtheusertochangeanyofthenumbers.Adecision-makermaybeallowedtomodifytheplanorscheduleinanotherinterfacewhichthenautomaticallychangesthevaluesoftheperformancemeasures,butausermaynotchangeperformancemeasuresdirectly.Userinterfacesfordatabasemodulesoftentakeafairlyconventionalformandmaybedeterminedbytheparticulardatabasepackageused.Thesein-terfacesmustallowforsomeuserinteraction,becausedatasuchasduedatesoftenhavetobechangedduringaplanningorschedulingsession.Thereareoftenanumberofinterfacesthatexhibitgeneraldataconcerningtheplantorenterprise.Examplesofsuchinterfacesare:(i)theplantlayoutinterface,(ii)theresourcecalendarinterface,and(iii)theroutingtableinterface.
14.5UserInterfacesandInteractiveOptimization359Theplantlayoutinterfacemaydepictgraphicallytheworkcentersandmachinesinaplantaswellasthepossibleroutesbetweentheworkcenters.Theresourcecalendardisplaysshiftschedules,holidaysandpreventivemain-tenanceschedulesofthemachines.Inthisinterfacetheusercanassignshiftsandscheduletheservicingoftheresources.Theroutingtabletypicallymayshowstaticdataassociatedwiththejobs.Itspecifiesthemachinesand/ortheoperatorswhocanprocessaparticularjoborjobtype.Themodulethatgeneratestheplansorschedulesmayprovidetheuserwithanumberofcomputationalproceduresandalgorithms.Suchalibraryofprocedureswithinthismodulewillrequireitsownuserinterface,enablingtheusertoselecttheappropriatealgorithmorevendesignanentirelynewprocedure.Userinterfacesthatdisplayinformationregardingtheplansorschedulescantakemanydifferentforms.Interfacesforadjustingormanipulatingtheplansorschedulesbasicallydeterminethecharacterofthesystem,asthesearetheonesusedmostextensively.Thevariousformsofinterfacesforma-nipulatingsolutionsdependonthelevelofdetailaswellasontheplanninghorizonbeingconsidered.Inwhatfollowsfoursuchinterfacesaredescribedinmoredetail,namely:(i)theGanttChartinterface,(ii)theDispatchListinterface,(iii)theCapacityBucketsinterface,and(iv)theThroughputDiagraminterface.Thefirst,andprobablymostpopular,formofschedulemanipulationin-terfaceistheGanttchart(seeFigure14.4).TheGanttchartistheusualhorizontalbarchart,withthex-axisrepresentingthetimeandthey-axis,thevariousmachines.Acolorand/orpatterncodemaybeusedtoindi-cateacharacteristicoranattributeofthecorrespondingjob.Forexam-ple,jobsthatarecompletedaftertheirduedateunderthecurrentsched-ulemaybecoloredred.TheGanttchartusuallyhasanumberofscrollcapabilitiesthatallowtheusertogobackandforthintimeorfocusonparticularmachines,andisusuallymousedriven.Iftheuserisnoten-tirelysatisfiedwiththegeneratedschedule,hemaywishtoperformsomemanipulationsonhisown.Withthemouse,theusercan“clickanddrag”anoperationfromonepositiontoanother.Providingtheinterfacewithaclick,drag,anddropcapabilityisnotatrivialtaskforthefollowingrea-son:afterchangingthepositionofaparticularoperationonamachine,otheroperationsonthatmachinemayhavetobepushedeitherforwardorbackwardintimetomaintainfeasibility.Thefactthatotheroperationshavetobeprocessedatdifferenttimesmayhaveaneffectonthesched-ulesofothermachines.Thisisoftenreferredtoascascadingorpropaga-tioneffects.Aftertheuserhasrepositionedanoperationofajob,thesys-temmaycallareoptimizationprocedurethatisembeddedintheplanningorschedulingenginetodealwiththecascadingeffectsinaproperman-ner.
36014SystemsDesignandImplementationFig.14.4.GanttChartInterfaceExample14.5.1(CascadingEffectsandReoptimization).Considerathreemachineflowshopwithunlimitedstoragespacebetweenthesuccessivemachinesandthereforenoblocking.Theobjectiveistominimizethetotalweightedtardiness.Consideraschedulewith4jobsasdepictedbytheGanttchartinFigure14.5.a.Iftheuserswapsjobs2and3onmachine1,whilekeepingtheorderonthetwosubsequentmachinesthesame,theresultingschedule,becauseofcascadingeffects,takestheformdepictedinFigure14.5.b.Ifthesystemhasreoptimizationalgorithmsatitsdisposal,theusermaydecidetoreoptimizetheoperationsonmachines2and3,whilekeepingthesequenceonmachine1frozen.AreoptimizationalgorithmthenmaygeneratethescheduledepictedinFigure14.5.c.Toobtainappropriatejobsequencesformachines2and3,thereoptimizationalgorithmhastosolveaninstanceofthetwomachineflowshopwiththejobssubjecttogivenreleasedatesatthefirstmachine.Ganttchartsdohavedisadvantages,especiallywhentherearemanyjobsandmachines.Itmaybehardtorecognizewhichbarorrectanglecorrespondstowhichjob.Asspaceonthescreen(orontheprintout)isratherlimited,itishardtoattachtexttoeachbar.Ganttchartinterfacesusuallyprovidethe
14.5UserInterfacesandInteractiveOptimization361Machine 1Machine 2Machine 31 341334341222Machine 1Machine 2Machine 31241324241333Machine 1Machine 2Machine 3134121 342234(a)(b)(c)Fig.14.5.Cascadingandreoptimizationafterswap:(a)originalschedule,(b)cas-cadingeffectsafterswapofjobsonmachine1,(c)scheduleafterreoptimizationofmachines2and3capabilitytoclickonagivenbarandopenawindowthatdisplaysdetaileddataregardingthecorrespondingjob.SomeGanttchartsalsohaveafiltercapability,wheretheusermayspecifythejob(s)thatshouldbeexposedontheGanttchartwhiledisregardingallothers.TheGanttchartinterfacedepictedinFigure14.4isfromtheLEKINsystemdescribedinChapter5.Thesecondformofuserinterfacedisplayingscheduleinformationisthedispatch-listinterface(seeFigure14.6).Schedulersoftenwanttoseealistofthejobstobeprocessedoneachmachineintheorderinwhichtheyaretobeprocessed.Withthistypeofdisplayschedulersalsowanttohaveediting
36214SystemsDesignandImplementationFig.14.6.Dispatch-listInterfacecapabilitiessotheycanchangethesequenceinwhichjobsareprocessedonamachineormoveajobfromonemachinetoanother.ThissortofinterfacedoesnothavethedisadvantageoftheGanttchart,sincethejobsarelistedwiththeirjobnumbersandtheschedulerknowsexactlywhereeachjobisinasequence.Iftheschedulerwouldliketoseemoreattributesofthejobstobelisted(e.g.,processingtime,duedate,completiontimeunderthecurrentschedule,andsoon),thenmorecolumnscanbeaddednexttothejobnum-bercolumn,eachonewithaparticularattribute.Thedisadvantageofthedispatch-listinterfaceisthattheschedulerdoesnothaveaclearviewoftheschedulerelativetotime.Theusermaynotseeimmediatelywhichjobsaregoingtobelate,whichmachineisidlemostofthetime,etc.Thedispatch-listinterfaceinFigure14.6isalsofromtheLEKINsystem.Thethirdformofuserinterfaceisthecapacitybucketsinterface.Thetimeaxisispartitionedintoanumberoftimeslotsorbuckets.Bucketsmaycorrespondtoeitherdays,weeksormonths.Foreachmachinetheprocessingcapacityofabucketisknown.Thecreationofplansorschedulesmayincertainenvironmentsbeaccomplishedbyassigningjobstomachinesingiventimesegments.Aftersuchassignmentsaremade,thecapacitybucketsinterface
14.5UserInterfacesandInteractiveOptimization363displaysforeachmachinethepercentageofthecapacityutilizedineachtimesegment.Ifthedecision-makerseesthatamachineisoverutilizedinagiventimeperiod,heknowsthatsomejobsinthecorrespondingbuckethavetoberescheduled.Thecapacitybucketsinterfacecontrasts,inasense,withtheGanttchartinterface.AGanttchartindicatesthenumberoflatejobsaswellastheirrespectivetardinesses.Thenumberoflatejobsandthetotalamountoftardinessesgiveanindicationofthedeficiencyincapacity.TheGanttchartisthusagoodindicatoroftheavailablecapacityintheshortterm(daysorweeks)whentherearealimitednumberofjobs(twentyorthirty).Capacitybucketsareusefulwhentheschedulerisperformingmediumorlongtermplanning.Thebucketsizemaybeeitheraweekoramonthandthetotalperiodcoveredthreeorfourmonths.Capacitybucketsare,ofcourse,acruderformofinformationastheydonotgiveanindicationofwhichjobsarecompletedontimeandwhichonesarecompletedlate.Thefourthformofuserinterfaceistheinput-outputdiagramorthroughputdiagraminterface,whichareoftenofinterestwhentheproductionismadetostock.Thesediagramsdescribethetotalamountofordersreceived,theto-talamountproducedandthetotalamountshipped,cumulativelyovertime.Thedifference,atanypointintime,betweenthefirsttwocurvesisthetotalamountoforderswaitingforprocessingandthedifferencebetweenthesecondandthethirdcurvesequalsthetotalamountoffinishedgoodsininventory.Thistypeofinterfacespecifiesneitherthenumberoflatejobsnortheirrespec-tivetardinesses.ItdoesprovidetheuserwithinformationregardingmachineutilizationandWork-In-Process(WIP).Clearly,thedifferentuserinterfacesforthedisplayofinformationregard-ingplansorscheduleshavetobestronglylinkedwithoneanother.WhenausermakeschangesineithertheGanttchartinterfaceorthedispatch-listinterface,thedynamicdatamaychangeconsiderablybecauseofcascadingef-fectsorthereoptimizationprocess.Changesmadeinoneinterface,ofcourse,havetobeshownimmediatelyintheotherinterfacesaswell.Userinterfacesthatdisplayinformationregardingtheplansorscheduleshavetobelinkedtootherinterfacesaswell,e.g.,databasemanagementinter-facesandinterfacesofaplanningorschedulingengine.Forexample,ausermaymodifyanexistingscheduleintheGanttchartinterfacebyclicking,drag-ging,anddropping;thenhemaywanttofreezecertainjobsintheirrespectivepositions.Afterdoingso,hemaywanttoreoptimizetheremaining(unfrozen)jobsusinganalgorithmintheschedulingengine.ThesealgorithmsaresimilartothealgorithmsdescribedinPartsIIandIIIforsituationswheremachinesarenotavailableduringgiventimeperiods(becauseofbreakdownsorotherreasons).Theinterfacesthatallowtheusertomanipulatetheplansorsched-uleshavetobe,therefore,stronglylinkedwiththeinterfacesforalgorithmselection.Userinterfacesmayalsohaveaseparatewindowthatdisplaysthevaluesofallrelevantperformancemeasures.Iftheuserhasmadeachangeinaplanorschedulethevaluesbeforeandafterthechangemaybedisplayed.
36414SystemsDesignandImplementationTypically,performancemeasuresaredisplayedinplaintextformat.However,moresophisticatedgraphicaldisplaysmayalsobeused.Someuserinterfacesaresophisticatedenoughtoallowtheusertosplitajobintoanumberofsmallersegmentsandscheduleeachoftheseseparately.Splittinganoperationisequivalentto(possiblymultiple)preemptions.Themoresophisticateduserinterfacesalsoallowdifferentoperationsofthesamejobtooverlapintime.Inpractice,thismayoccurinmanysettings.Forexample,ajobmaystartatadownstreammachineofaflowshopbeforeithascompleteditsprocessingatanupstreammachine.Thisoccurswhenajobrepresentsalargebatchofidenticalitems.Beforetheentirebatchhasbeencompletedattheupstreammachine,partsofthebatchmayalreadyhavebeentransportedtothenextmachineandmayalreadyhavestartedtheirprocessingthere.14.6GenericSystemsvs.Application-SpecificSystemsDozensofsoftwarehouseshavedevelopedsystemswhichtheyclaimcanbeimplementedinmanydifferentindustrialsettingsafteronlyminormodifica-tions.Itoftenturnsoutthattheeffortinvolvedincustomizingsuchsystemsisquitesubstantial.Thecodedevelopedforthecustomizationofthesystemmayturnouttobemorethanhalfthetotalcodeofthefinalversionofthesys-tem.However,somesystemshaveverysophisticatedconfigurationsthatallowthemtobetailoredtodifferenttypesofindustrieswithoutmuchprogram-mingeffort.Thesesystemsarehighlymodularandhaveanedgewithregardtoadjustmentstospecificrequirements.Agenericsystem,ifitishighlymod-ular,canbechangedtofitaspecificenvironmentbyaddingspecificmodules,e.g.,tailor-madeschedulingalgorithms.Expertscandevelopsuchalgorithmsandthegenericplanningandschedulingsoftwaresuppliesstandardinterfacesor“hooks”thatallowtheintegrationofspecialfunctionsinthepackage.Thisconceptallowstheexpertstoconcentrateontheplanningorschedulingprob-lem,whilethegenericsoftwarepackagesuppliesthefunctionalitiesthatarelessspecific,e.g.,userinterfaces,datamanagement,standardplanningandschedulingalgorithmsforlesscomplexareas,andsoon.Genericsystemsmaybebuilteitherontopofacommercialdatabasesys-tem,suchasSybaseorOracle,oraproprietarydatabasesystemdevelopedspecificallyfortheplanningorschedulingsystem.Genericsystemsusepro-cessingdatasimilartothedatapresentedintheframeworksdescribedinChapters2and3.However,theframeworkinsuchasystemmaybesome-whatmoreelaboratethantheframeworkspresentedinChapters2and3.Forexample,thedatabasemayallowforanalphanumericordernumber,thatreferstothenameofacustomer.Theordernumberthenrelatestoseveraljobs,eachonewithitsownprocessingtime(whichoftenmaybereferredtoasthequantityofabatch)andaroutingvectorthatdeterminestheprece-denceconstraintstheoperationsaresubjectto.Theordernumberhasits
14.6GenericSystemsvs.Application-SpecificSystems365ownduedate(committedshippingdate),weight(priorityfactor)andreleasedate(whichmaybedeterminedbyaMaterialRequirementsPlanning(MRP)systemconnectedtotheplanningorschedulingsystem).Thesystemmayin-cludeproceduresthattranslatetheduedateoftheorderintoduedatesforthedifferentjobsatthevariousworkcenters.Also,theweightsofthedifferentjobsbelongingtoanordermaynotbeexactlyequaltotheweightoftheorderitself.Theweightsofthedifferentjobsmaybeafunctionoftheamountofvaluealreadyaddedtotheproduct.Theweightofthelastjobpertainingtoanordermaybelargerthantheweightofthefirstjobpertainingtothatorder.Thewaythemachineorresourceenvironmentisrepresentedinthedata-baseisalsosomewhatmoreelaboratethanthewayitisdescribedinChapters2and3.Forexample,asystemtypicallyallowsaspecificationofworkcentersand,withineachworkcenter,aspecificationofmachines.Mostgenericplanningandschedulingsystemshaveroutinesforgenerat-inga“first”planorschedulefortheuser.Ofcourse,suchaninitialsolutionrarelysatisfiestheuser.Thatiswhyplanningandschedulingsystemsoftenhaveelaborateuserinterfacesthatallowtheusertomanuallymodifyanex-istingplanorschedule.Theautomatedplanningandschedulingcapabilitiesgenerallyconsistofanumberofdifferentdispatchingrulesthatarebasicallysortingroutines.Theserulesareusuallythesameasthepriorityrulesdis-cussedinthepreviouschapters(SPT,LPT,WSPT,EDDandsoon).Somegenericsystemsrelyonmoreelaborateprocedures,suchasforwardloadingorbackwardloading.Forwardloadingimpliesthatthejobsareinsertedoneatthetimestartingatthebeginningoftheschedule,thatis,atthecurrenttime.Backwardloadingimpliesthatthescheduleisgeneratedstartingfromthebackoftheschedule,thatis,fromtheduedates,workingitswaytowardsthecurrenttime(againinsertingonejobatthetime).Theseinsertions,ei-therforwardorbackwardintime,aredoneaccordingtosomepriorityrule.Someofthemoresophisticatedautomatedproceduresfirstidentifythebottle-neckworkcenter(s)ormachine(s);theycomputetimewindowsduringwhichjobshavetobeprocessedonthesemachinesandthentheyschedulethejobsonthesemachinesthroughsomealgorithmicprocedure.Afterthebottlenecksarescheduled,theprocedureschedulestheremainingmachinesthrougheitherforwardloadingorbackwardloading.AlmostallgenericplanningandschedulingsystemshaveuserinterfacesthatincludeGanttchartsandenabletheusertomanipulatethesolutionsmanually.However,theseGanttchartinterfacesarenotalwaysperfect.Forexample,mostofthemdonottakeintoaccountthecascadingandpropaga-tioneffectsreferredtointheprevioussection.Theymaydosomeautomaticreschedulingonthemachineorworkcenterwherethedecisionmakerhasmadeachange,buttheyusuallydonotadapttheschedulesonothermachinesorworkcenterstothischange.Thesolutionsgeneratedmayattimesbeinfeasi-ble.Somesystemsgivetheuserawarningwhen,afterthemodifications,theresultingplanorscheduleturnsouttobeinfeasible.
36614SystemsDesignandImplementationBesidestheGanttchartinterface,mostsystemshaveatleastoneothertypeofinterfacethateitherdisplaystheactualplanorscheduleorprovidesimportantdatathatisrelated.Thesecondinterfaceistypicallyoneofthosementionedintheprevioussection.Genericsystemsusuallyhavefairlyelaboratereportgenerators,thatprintouttheplanorschedulewithalphanumericcharacters;suchprintoutscanbedonefastandonaninexpensiveprinter.Theprintoutmaythenresemblewhatisdisplayed,forexample,inthedispatch-listinterfacedescribedintheprevioussection.Itispossibletolistthejobsintheorderinwhichtheywillbeprocessedataparticularmachineorworkcenter.Besidesthejobnumber,otherrelevantjobdatamaybeprintedoutaswell.TherearealsosystemsthatprintoutentireGanttcharts.ButGanttchartshavethedisadvantagementionedbefore,namelythatitmaynotbeimmediatelyobviouswhichrectangleorbarcorrespondstowhichjob.Usuallythebarsaretoosmalltoappendanyinformationto.Genericsystemshaveanumberofadvantagesoverapplication-specificsystems.Iftheschedulingproblemisafairlystandardoneandonlyminorcustomizationofagenericsystemsuffices,thenthisoptionisusuallylessexpensivethandevelopinganapplication-specificsystemfromscratch.Anadditionaladvantageisthatanestablishedcompanywillmaintainthesystem.Ontheotherhand,mostsoftwarehousesthatdevelopplanningorschedulingsystemsdonotprovidethesourcecode.Thismakestheuserofthesystemdependentonthesoftwarehouseevenforveryminorchanges.Inmanyinstancesgenericsystemsaresimplynotsuitableandapplication-specificsystems(ormodules)havetobedeveloped.Thereareseveralgoodreasonsfordevelopingapplication-specificsystems.Onereasonmaybethattheplanningorschedulingproblemissimplysolarge(becauseofthenum-berofmachines,jobs,orattributes)thataPC-basedgenericsystemsimplywouldnotbeabletohandleit.Thedatabasesmaybeverylargeandtherequiredinterfacebetweentheshopfloorcontrolsystemandtheplanningorschedulingsystemmaybeofakindwhichagenericsystemcannothandle.Anexampleofanenvironmentwherethisisoftenthecaseissemiconductormanufacturing.Asecondreasontooptforanapplication-specificsystemisthattheen-vironmentmayhavesomanyidiosyncrasiesthatnogenericsystemcanbemodifiedinsuchawaythatitcanaddresstheproblemsatisfactorily.Theprocessingenvironmentmayhavecertainrestrictionsorconstraintsthataredifficulttoattachtoorbuildintoagenericsystem.Forexample,certainma-chinesataworkcentermayhavetostartwiththeprocessingofdifferentjobsatthesametime(foronereasonoranother)oragroupofmachinesmayhavetosometimesactasasinglemachineand,atothertimes,asseparatema-chines.Theorderportfoliomayalsohavemanyidiosyncrasies.Thatis,theremaybeafairlycommonmachineenvironmentusedinafairlystandardway(thatwouldfitnicelyintoagenericsystem),butwithtoomanyexceptionsontherulesasfarasthejobsareconcerned.Codinginthespecialsituations
14.7ImplementationandMaintenanceIssues367representssuchalargeamountofworkthatitmaybeadvisabletobuildasystemfromscratch.Athirdreasonfordevelopinganapplication-specificsystemisthattheusermayinsistonhavingthesourcecodeandbeabletomaintainthesystemwithinhisownorganization.Animportantadvantageofanapplication-specificsystemisthatmanip-ulatingasolutionisusuallyconsiderablyeasierthanwithagenericsystem.14.7ImplementationandMaintenanceIssuesDuringthelasttwodecadesalargenumberofplanningandschedulingsys-temshavebeendeveloped,andmanymoreareunderdevelopment.Thesedevelopmentshavemadeitclearthatacertainproportionofthetheoreticalresearchdoneoverthelastcoupleofdecadesisofverylimiteduseinrealworldapplications.Fortunately,thesystemdevelopmentthatisgoingoninindustryiscurrentlyencouragingtheoreticalresearcherstotackleplanningandschedulingproblemsthataremorerelevanttotherealworld.Atvari-ousacademicinstitutionsinEurope,JapanandNorthAmerica,researchisfocusingonthedevelopmentofalgorithmsaswellasonthedevelopmentofsystems;significanteffortsarebeingmadeinintegratingthesedevelopments.Overthelasttwodecadesmanycompanieshavemadelargeinvestmentsinthedevelopmentandimplementationofplanningandschedulingsystems.However,notallthesystemsdevelopedorinstalledappeartobeusedonaregularbasis.Systems,afterbeingimplemented,oftenremaininuseonlyforalimitedtime;afterawhiletheymaybeignoredaltogether.Inthosesituationswherethesystemsareinuseonamoreorlessper-manentbasis,thegeneralfeelingisthattheoperationsdorunsmoother.Asystemthatisinplaceoftendoesnotreducethetimethedecision-makerspendsonplanningandscheduling.However,asystemusuallydoesenabletheusertoproducebettersolutions.ThroughaninteractiveGraphicsUserInterface(GUI)auserisoftenabletocomparedifferentsolutionsandmoni-torthevariousperformancemeasures.Thereareotherreasonsforsmootheroperationsbesidessimplybetterplansandschedules.Aplanningorschedul-ingsystemimposesacertain”discipline”ontheoperations.Therearenowcompellingreasonsforkeepinganaccuratedatabase.Plansandschedulesareeitherprintedoutneatlyorvisibleonmonitors.Thisapparentlyhasaneffectonpeople,encouragingthemtoactuallyevendotheirjobsaccordingtotheplanorschedule.Thesystemdesignershouldbeawareofthereasonswhysomesystemshaveneverbeenimplementedorareneverused.Insomecases,databasesarenotsufficientlyaccurateandtheteamimplementingthesystemdoesnothavethepatienceortimetoimprovethedatabase(thepersonsresponsibleforthedatabasemaybedifferentfromthepeopleinstallingtheschedulingsystem).Inothercases,thewayinwhichworkers’productivityismeasuredisnot
36814SystemsDesignandImplementationinagreementwiththeperformancecriteriathesystemisbasedupon.Userinterfacesmaynotpermittheuserofthesystemtoreschedulesufficientlyfastinthecaseofunexpectedevents.Proceduresthatenablereschedulingwhenthemainuserisabsent(forexample,ifsomethingunexpectedhappensduringthirdshift)maynotbeinplace.Finally,systemsmaynotbegivensufficienttimeto”settle”or”stabilize”intheirenvironment(thismayrequiremanymonths,ifnotyears).Evenifasystemgetsimplementedandused,thedurationduringwhichitremainsinusemaybelimited.Everysooften,theorganizationmaychangedrasticallyandthesystemisnotflexibleenoughtoprovidegoodplansorschedulesforthenewenvironment.Evenachangeinamanagermayderailasystem.Insummary,thefollowingpointscouldbetakenintoconsiderationwhendesigning,developingandimplementingasystem.1.Visualizehowtheoperatingenvironmentwillevolveoverthelifetimeofthesystembeforethedesignprocessactuallystarts.2.Getallthepeopleaffectedbythesysteminvolvedinthedesignprocess.Thedevelopmentprocesshastobeateameffortandallinvolvedhavetoapprovethedesignspecifications.3.Determinewhichpartofthesystemcanbehandledbyoff-the-shelfsoft-ware.Usinganappropriatecommercialcodespeedsupthedevelopmentprocessconsiderably.4.Keepthedesignofthesoftwaremodular.Thisisnecessarynotonlytofacilitatetheentireprogrammingeffort,butalsotofacilitatechangesinthesystemafteritsimplementation.5.Maketheobjectivesofthealgorithmsembeddedinthesystemconsistentwiththeperformancemeasuresbywhichpeoplewhomustactaccordingtotheplansorschedulesarebeingjudged.6.Donottakethedataintegrityofthedatabaseforgranted.Thesystemhastobeabletodealwithfaultyormissingdataandprovidethenecessarysafeguards.7.Capitalizeonpotentialsidebenefitsofthesystem,e.g.,spin-offreportsfordistributiontokeypeople.Thisenlargesthesupportersbaseofthesystem.8.Makeprovisionstoensureeasyrescheduling,notonlybythemainplannerorschedulerbutalsobyothers,incasethemainuserisabsent.9.Keepinmindthattheinstallmentofthesystemrequirespatience.Itmaytakemonthsorevenyearsbeforethesystemrunssmoothly.Thisperiodshouldbeaperiodofcontinuousimprovement.10.Donotunderestimatethenecessarymaintenanceofthesystemafterin-stallation.Theeffortrequiredtokeepthesysteminuseonaregularbasisisconsiderable.Itappearsthatinthedecadetocome,anevenlargereffortwillbemadeinthedesign,developmentandimplementationofplanningandscheduling
Exercises369systemsandthatsuchsystemswillplayanimportantroleinComputerInte-gratedManufacturing.Exercises14.1.Considerajobshopwithmachinesinparallelateachworkcenter(i.e.,aflexiblejobshop).Therearehardconstraintsaswellassoftconstraintsthatplayaroleintheschedulingofthemachines.Moremachinesmaybeinstalledinthenearfuture.Theschedulingprocessdoesnothavetobedoneinrealtime,butcanbedoneovernight.Describetheadvantagesanddisadvantagesofanalgorithmicapproachandofaknowledge-basedapproach.14.2.Considerafactorywithasinglemachinewithsequencedependentsetuptimesandhardduedates.Itdoesnotappearthatchangesintheenvironmentareimminentinthenearfuture.Schedulingandreschedulinghastobedoneinrealtime.(a)Listtheadvantagesanddisadvantagesofanalgorithmicapproachandofaknowledge-basedapproach.(b)Listtheadvantagesanddisadvantagesofacommercialsystemandofanapplication-specificsystem.14.3.Designaschedulegenerationmodulethatisbasedonacompositedis-patchingruleforaparallelmachineenvironmentwiththejobssubjecttosequencedependentsetuptimes.JobjhasreleasedaterjandmayonlybeprocessedonamachinethatbelongstoagivensetMj.Therearethreeob-jectives,namelywjTj,CmaxandLmax.Eachobjectivehasitsownweightandtheweightsaretimedependent;everytimetheschedulerusesthesystemheputsintherelativeweightsofthevariousobjectives.Designthecompos-itedispatchingruleandexplainhowthescalingparametersdependontherelativeweightsoftheobjectives.14.4.Considerthefollowingthreemeasuresofmachinecongestionoveragiventimeperiod.(i)thenumberoflatejobsduringtheperiod;(ii)theaveragenumberofjobswaitinginqueueduringthegivenperiod.(iii)theaveragetimeajobhastowaitinqueueduringtheperiod.Howdoestheselectionofcongestionmeasuredependontheobjectivetobeminimized?14.5.Considerthefollowingschedulingalternatives:(i)forwardloading(startingfromthecurrenttime);(ii)backwardloading(startingfromtheduedates);(iii)schedulingfromthebottleneckstagefirst.Howdoestheselectionofoneofthethreealternativesdependonthefollowingfactors:
37014SystemsDesignandImplementation(i)degreeofuncertaintyinthesystem.(ii)balancedoperations(notonespecificstageisabottleneck).(iii)duedatetightness.14.6.ConsidertheATCrule.TheKfactorisusuallydeterminedasafunctionoftheduedatetightnessfactorθ1andtheduedaterangefactorθ2.However,theprocessusuallyrequiresextensivesimulation.Designalearningmechanismthatrefinesthefunctionfthatmapsθ1andθ2intoKduringtheregular(possiblydaily)useofthesystem’sschedulegenerator.14.7.Consideraninteractiveschedulingsystemwithauserinterfaceforschedulemanipulationthatallows“freezing”ofjobs.Thatis,theschedulercanclickonajobandfreezethejobinacertainposition.Theotherjobshavetobescheduledaroundthefrozenjobs.Freezingcanbedonewithtol-erances,sothatintheoptimizationprocessoftheremainingjobsthefrozenjobscanbemovedalittlebit.Thisfacilitatestheschedulingoftheunfrozenjobs.Considerasystemthatallowsfreezingofjobswithspecifiedtolerancesandshowthatfreezinginanenvironmentthatdoesnotallowpreemptionsrequirestolerancesofatleasthalfthemaximumprocessingtimeineitherdirectioninordertoavoidmachineidletimes.14.8.Consideraninteractiveschedulingsystemwithauserinterfacethatonlyallowsforfreezingofjobswithno(zero)tolerances.(a)Showthatinanonpreemptiveenvironmentthemachineidletimescausedbyfrozenjobsarealwayslessthanthemaximumprocessingtime.(b)Describehowprocedurescanbedesignedthatminimizeinsuchascenariomachineidletimesinconjunctionwithotherobjectives,suchasthetotalcompletiontime.14.9.Considerauserinterfaceofaninteractiveschedulingsystemforabankofparallelmachines.Assumethatthereoptimizationalgorithmsinthesystemaredesignedinsuchawaythattheyoptimizeeachmachineseparatelywhiletheykeepthecurrentassignmentofjobstomachinesunchanged.Amoveofajob(withthemouse)issaidtobereversibleifthemove,followedbythereoptimizationprocedure,followedbythereversemove,followedoncemorebythereoptimizationprocedure,resultsintheoriginalschedule.Supposeajobismovedwiththemousefromonemachinetoanother.Showthatsuchamoveisreversibleifthereoptimizationalgorithmminimizesthetotalcompletiontime.Showthatthesameistrueifthereoptimizationalgorithmminimizesthesumoftheweightedtardinesses.14.10.Considerthesamescenarioasinthepreviousexercise.Showthatwiththetypeofreoptimizationalgorithmsdescribedinthepreviousexercisemovesthattakejobsfromonemachineandputthemonanotherarecommutative.Thatis,thefinalscheduledoesnotdependonthesequenceinwhichthemovesaredone,evenifallmachinesarereoptimizedaftereachmove.
CommentsandReferences371CommentsandReferencesManypapersandbookshavebeenwrittenonthevariousaspectsofproductioninformationsystems;see,forexample,Gaylord(1987),Scheer(1988)andPimentel(1990).Withregardtotheissuesconcerningtheoveralldevelopmentofplanningandschedulingsystemsarelativelylargenumberofpapershavebeenwritten,ofteninproceedingsofconferences,e.g.,Oliff(1988),KarwanandSweigart(1989),Interrante(1993).SeealsoKanetandAdelsberger(1987),KusiakandChen(1988),Solberg(1989),AdelsbergerandKanet(1991),Pinedo,SamroengrajaandYen(1994),andPinedoandYen(1997).Forresearchfocusingspecificallyonknowledge-basedsystems,seeSmith,FoxandOw(1986),ShawandWhinston(1989),Atabakhsh(1991),NoronhaandSarma(1991),Lefrancois,JobinandMontreuil(1992)andSmith(1992,1994).Forworkonthedesignanddevelopmentofuserinterfacesforplanningandschedulingsystems,seeKempf(1989)andWoernerandBiefeld(1993).
Chapter15AdvancedConceptsinSystemsDesign15.1Introduction……………………………37315.2RobustnessandReactiveDecisionMaking…..37415.3MachineLearningMechanisms…………….37915.4DesignofPlanningandSchedulingEnginesandAlgorithmLibraries………………….38515.5ReconfigurableSystems…………………..38815.6Web-BasedPlanningandSchedulingSystems..39015.7Discussion……………………………..39315.1IntroductionThischapterfocusesonanumberofissuesthathavecomeupinrecentyearsinthedesign,development,andimplementationofplanningandschedulingsystems.Thenextsectiondiscussesissuesconcerninguncertainty,robustnessandreactivedecisionmaking.Inpractice,plansorschedulesoftenhavetobechangedbecauseofrandomevents.Themorerobusttheoriginalplanorscheduleis,theeasierthereplanningorreschedulingprocessis.Thissectionfocusesonthegenerationofrobustplansandschedulesaswellasthemea-surementoftheirrobustness.Thethirdsectionconsidersmachinelearningmechanisms.Asystemcannotconsistentlygenerategoodsolutionsthataretothelikingoftheuser.Thedecision-makeroftenhastotweaktheplanorschedulegeneratedbythesysteminordertomakeitusable.Awell-designedsystemcanlearnfromadjustmentsmadebytheuserinthepast;themech-anismthatallowsthesystemtodosoistypicallyreferredtoasalearningmechanism.Thefourthsectionfocusesonthedesignofplanningandschedul-ingengines.Anengineoftencontainsalibraryofalgorithmsandroutines.Oneproceduremaybemoreappropriateforonetypeofinstanceordata© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_15,373
37415AdvancedConceptsinSystemsDesignset,whileanotherproceduremaybemoreappropriateforanothertypeofin-stance.Theusershouldbeabletoselect,foreachinstance,whichproceduretoapply.Itmayevenbethecasethatauserwouldliketotackleaninstanceusingacombinationofvariousprocedures.Thisfourthsectiondiscusseshowaplanningorschedulingengineshouldbedesignedinordertoenabletheusertoadaptandcombinealgorithmsinordertoachievemaximumeffectiveness.Thefifthsectionfocusesonreconfigurablesystems.Experiencehasshownthatthedevelopmentandimplementationofsystemsisverytimeconsumingandcostly.Inordertoreducethecosts,effortsshouldbemadetomaintainahighdegreeofmodularityinthedesignofthesystem.Ifthemodulesarewelldesignedandsufficientlyflexible,theycanbeusedoverandoveragainwithoutanymajorchanges.Thesixthsectionfocusesonthedesignaspectsofweb-basedplanningandschedulingsystems.Thissectiondiscussestheeffectsofnetworkingonthedesignofsuchsystems.Theseventhandlastsectiondiscussesanumberofotherissuesandpresentsaviewonhowplanningandschedulingsystemsmayevolveinthefuture.15.2RobustnessandReactiveDecisionMakingInpractice,itoftenhappensthatsoonafteraplanorschedulehasbeengen-erated,anunexpectedeventhappensthatforcesthedecision-makertomakechanges.Suchaneventmay,forexample,beamachinebreakdownorarushjobthatsuddenlyhastobeinserted.Manyplannersandschedulersbelievethatinpractice,mostofthetime,thedecisionmakingprocessisareactiveprocess.Inareactiveprocess,theplannerorschedulertriestoaccomplishanumberofobjectives.Hetriestoaccomodatetheoriginalobjectives,andalsotriestomakethenewplanorschedulelook,asmuchaspossible,liketheoriginaloneinordertominimizeconfusion.Theremainingpartofthissectionfocusesprimarilyonreactivedecisionmakinginshorttermschedulingprocesses.Thenumberofrandomeventsthatcanoccurinashorttermmay,incertainenvironments,beveryhigh.Reschedulingisinmanyenvironmentsawayoflife.Onewayofdoingthereschedulingistoputalltheoperationsnotyetstartedbackinthehopper,andgenerateanewschedulefromscratchwhiletakingintoaccountthedisruptionsthatjustoccurred.Thedangeristhatthenewschedulemaybecompletelydifferentfromtheoriginalschedule,andabigdifferencemaycausesomeconfusion.Ifthedisruptionisminor,e.g.,thearrivalofjustoneunexpectedjob,thenasimplechangemaysuffice.Forexample,theschedulermayinserttheunex-pectedarrivalinthecurrentscheduleinsuchawaythatthetotaladditionalsetupisminimizedandnootherhighpriorityjobisdelayed.Amajordisrup-tion,likethebreakdownofanimportantmachine,oftenrequiressubstantialchangesintheschedule.Ifamachinegoesdownforanextendedperiodof
15.2RobustnessandReactiveDecisionMaking375time,thentheentireworkloadallocatedtothatmachineoverthatperiodhastobetransferredtoothermachines.Thismaycauseextensivedelays.Anotherwayofdealingwiththereschedulingprocessistosomehowan-ticipatetherandomevents.Inordertodoso,itisnecessaryfortheoriginalscheduletoberobustsothatthechangesafteradisruptionareminimal.Schedulerobustnessisaconceptthatisnoteasytomeasureorevendefine.Supposethecompletiontimeofajobisdelayedbyδ(becauseofamachinebreakdownortheinsertionofarushjob).LetCj(δ)denotethenewcompletiontimeofjobj(i.e.,thenewtimeatwhichjobjleavesthesystem),assumingthesequencesofalltheoperationsonallthemachinesremainthesame.Ofcourse,thenewcompletiontimesofallthejobsareafunctionofδ.LetZdenotethevalueoftheobjectivefunctionbeforethedisruptionoccurredandletZ(δ)denotethenewvalueoftheobjectivefunction.SoZ(δ)−Zisthedifferenceduetothedisruption.OnemeasureofschedulerobustnessisZ(δ)−Zδ,whichisafunctionofδ.Forsmallvaluesofδtheratiomaybelowwhereasforlargervaluesofδitmaygetprogressivelyworse.Itistobeexpectedthatthisratioisincreasingconvexinδ.Amoreaccuratemeasureofrobustnesscanbeestablishedwhentheproba-bilitiesofcertaineventscanbeestimatedinadvance.Supposeaperturbationofarandomsize∆mayoccurandtheprobabilitytherandomvariable∆assumesthevalueδ,i.e.,P(∆=δ),canbeestimated.If∆canassumeonlyintegervalues,then∞δ=0Z(δ)−ZP(∆=δ)isanappropriatemeasurefortherobustness.Iftherandomvariable∆isacontinuousrandomvariablewithadensityfunctionf(δ),thenanappropriatemeasureis∞δ=0(Z(δ)−Z)f(δ)dδ.Inpractice,itmaybedifficulttomakeaprobabilisticassessmentofrandomperturbationsandonemaywanttohavemorepracticalmeasuresofrobust-ness.Forexample,onemeasurecouldbebasedontheamountofslackbetweenthecompletiontimesofthejobsandtheirrespectiveduedates.SoapossiblemeasurefortherobustnessofscheduleSisR(S)=nj=1wj(dj−Cj)wjdj.ThelargerR(S),themorerobusttheschedule.Maximizingthisparticularmeasureofrobustnessissomewhatsimilartomaximizingthetotalweightedearliness.
37615AdvancedConceptsinSystemsDesignWhenshouldadecision-makeroptforamorerobustschedule?Thismaydependontheprobabilityofadisruptionaswellasonhisorherabilitytoreschedule.Example15.2.1(MeasuresofRobustness).Considerasinglemachineandthreejobs.Thejobdataarepresentedinthetablebelow.jobs123pj101010dj102234wj1100100Theschedulethatminimizesthetotalweightedtardinessisschedule1,2,3withatotalweightedtardinessof0.Itisclearthatthisscheduleisnotthatro-bust,sincetwojobswithverylargeweightsarescheduledforcompletionveryclosetotheirrespectiveduedates.Supposethatimmediatelyafterschedule1,2,3hasbeenfixedadisruptionoccurs,i.e.,attime0+,andthemachinegoesdownforδ=10timeunits.Themachinecanstartprocessingthethreejobsattimet=10.Iftheoriginaljobsequence1,2,3hastobemaintained,thenthetotalweightedtardinessis1410.Themannerinwhichthetotalweightedtardinessofsequence1,2,3dependsonthevalueofδisdepictedinFigure15.1.Iftheoriginalscheduleis2,3,1,thenthetotalweightedtardiness,withnodisruptions,is20.However,ifadisruptiondoesoccurattime0+,thentheimpactisconsiderablylessseverethanwithschedule1,2,3.Ifδ=10,thenthetotalweightedtardinessis30.Thewaythetotalweightedtardinessundersequence2,3,1dependsonδisalsodepictedinFigure15.1.FromFigure15.1itisclearthatschedule2,3,1(eventhoughoriginallysuboptimal)ismorerobustthanschedule1,2,3.Underschedule1,2,3therobustnessisR(1,2,3)=nj=1wj(dj−Cj)wjdj=6005610=0.11,whereasR(2,3,1)=25805610=0.46.Soaccordingtothisparticularmeasureofrobustnessschedule2,3,1iscon-siderablymorerobust.Supposethatwithprobability0.01arushjobwithprocessingtime10arrivesattime0+andthatthedecision-makerisnotallowed,atthecom-pletionofthisrushjob,tochangetheoriginaljobsequence.Ifattheoutsethehadselectedschedule1,2,3,thenthetotalexpectedweightedtardinessis0×0.9+1410×0.1=141.
15.2RobustnessandReactiveDecisionMaking377(1, 2, 3)(2, 3, 1)200010001020wjTjΣδFig.15.1.IncreaseinObjectiveValueasaFunctionofDisruptionLevelIfhehadselectedschedule2,3,1,thenthetotalexpectedweightedtardinessis20×0.9+30×0.1=21.So,whenthereisa10%probabilityofadisruptionitisbettertogoforthemorerobustschedule.Evenifaschedulerisallowedtorescheduleafteradisruption,hestillmaynotchooseattime0aschedulethatisoptimalwithrespecttotheoriginaldata.Severalothermeasuresofrobustnesscanbedefined.Forexample,assumeagainthatthecompletionofonejobisdelayedbyδ.However,beforecom-putingtheeffectofthedisruptionontheobjective,eachmachinesequenceisreoptimizedseparately,i.e.,themachinesequencesarereoptimizedonebyoneonamachinebymachinebasis.Afterthisreoptimizationthedifferenceintheobjectivefunctioniscomputed.Themeasureofrobustnessisthenasimilarratioastheonedefinedabove.Theimpactofthedisruptionisnowhardertocompute,sincedifferentvaluesofδmayresultindifferentschedules.Thisratiois,ofcourse,lessthantheratiowithoutreoptimization.Anevenmorecomplicatedmeasureofrobustnessassumesthatafteradisruptionareopti-mizationisdoneonamoreglobalscaleratherthanonamachinebymachinebasis,e.g.,underthisassumptionadisruptionmaycauseanentirejobshop
37815AdvancedConceptsinSystemsDesigntobereoptimized.Othermeasuresofrobustnessmayevenallowpreemptionsinthereoptimizationprocess.Totallydifferentmeasuresofrobustnesscanbedefinedbasedontheca-pacityutilizationofthebottleneckmachines(i.e.,thepercentagesoftimethemachinesareutilized)andonthelevelsofWIPinventorythatarekeptinfrontofthesemachines.Howcanonegeneraterobustschedules?Onecanfollowvariousrulesinordertocreatesuchschedules,forexample,(i)insertidletimes,(ii)schedulelessflexiblejobsfirst,(iii)donotpostponetheprocessingofanyoperationunnecessarily,and(iv)keepalwaysanumberofjobswaitinginfrontofhighlyutilizedma-chines.Thefirstruleprescribestheinsertionofidleperiodsongivenresourcesatcertainpointsintime.Thisisequivalenttoschedulingthemachinesbelowcapacity.Thedurationsoftheidleperiodsaswellastheirtimingwithinthescheduledependontheexpectednatureofthedisruptions.Onecouldarguethattheidleperiodsinthebeginningoftheschedulemaybekeptshorterthantheidleperiodslaterintheschedule,sincetheprobabilityofaneventoccurringinthebeginningmaybesmallerthanlateron.Inpractice,someschedulersfollowarulewherebyatanypointintimeinthecurrentweekthemachinesareutilizedupto90%ofcapacity,inthenextweekupto80%andintheweekafterthatupto70%.However,onereasonforkeepingtheidleperiodsinthebeginningofthescheduleatthesamelengthmaybethefollowing:eventhoughtheprobabilityofadisruptionissmall,itsrelativeimpactismoreseverethanthatofadisruptionthatoccurslateronintheprocess.Thesecondrulesuggeststhatlessflexiblejobsshouldhaveahigherprioritythanmoreflexiblejobs.Ifadisruptionoccurs,thenthemoreflexiblejobsremaintobeprocessed.Theflexibilityofajobisdetermined,forexample,bythenumberofmachinesthatcandotheprocessing(e.g.,themachineeligibilityconstraintsdescribedinChapter2).However,theflexibilityofajobmayalsobedeterminedbythesetuptimestructure.Somejobsmayrequiresetupsthatdonotdependonthesequence.Otherjobsmayhavesequencedependentsetuptimesthatarehighlyvariable.Thesetuptimesareshortonlywhentheyfollowcertainotherjobs;otherwisethesetuptimesareverylong.Suchjobsareclearlylessflexible.Thethirdrulesuggeststhattheprocessingofajobshouldnotbepost-ponedunnecessarily.Fromthepointofviewofinventoryholdingcostsandearlinesspenalties,itisdesirabletostartoperationsaslateaspossible.Fromarobustnesspointofview,itmaybedesirabletostartoperationsasearlyaspossible.Sothereisatrade-offbetweenrobustnessandearlinesspenaltiesorinventoryholdingcosts.Thefourthruletriestomakesurethatabottleneckmachineneverstarvesbecauseofrandomeventsthatoccurupstream.Itmakessensetohavealways
15.3MachineLearningMechanisms379anumberofjobswaitingforprocessingatabottleneckmachine.Thereasonisthefollowing:ifnoinventoryiskeptinfrontofthebottleneckandthemachinefeedingthebottlenecksuddenlybreaksdown,thenthebottleneckmayhavetoremainidleandmaynotbeabletomakeupforthelosttimelateron.Example15.2.2(StarvationAvoidance).Consideratwomachineflowshopwith100identicaljobs.Eachjobhasaprocessingtimeof5timeunitsonmachine1andof10timeunitsonmachine2.Machine2isthereforethebottleneck.However,aftereachjobcompletiononmachine1,machine1mayhavetoundergoamaintenanceserviceforadurationof45timeunitsduringwhichitcannotdoanyprocessing.Theprobabilitythatsuchaserviceisrequiredis0.01.Theprimaryobjectiveistheminimizationofthemakespanandthesec-ondaryobjectiveistheaverageamountoftimeajobremainsinthesystem,i.e.,thetimeinbetweenthestartofajobonmachine1anditscompletiononmachine2(thissecondaryobjectiveisbasicallyequivalenttotheminimiza-tionoftheWork-In-Process).However,theweightoftheprimaryobjectiveis1000timestheweightofthesecondaryobjective.Becauseofthesecondaryobjectiveitdoesnotmakesensetoletmachine1processthe100jobsoneafteranotherandfinishthemallbytime500.Inanenvironmentinwhichmachine1neverrequiresanyservicing,theoptimalscheduleprocessesthejobsonmachine1withidletimesof5timeunitsinbetween.Inanenvironmentinwhichmachine1needsservicingwithagivenprobability,itisnecessarytohaveatalltimessomejobsreadyforprocessingonmachine2.Theoptimalscheduleistokeepconsistently5jobswaitingforprocessingonmachine2.Ifmachine1hastobeserviced,thenmachine2doesnotloseanytimeandthemakespandoesnotgoupunnecessarily.Thisexampleillustratesthetrade-offbetweencapacityutilizationandminimizationofWork-In-Process.Robustnessandreschedulinghaveastronginfluenceonthedesignoftheuserinterfacesandonthedesignoftheschedulingengine(multi-objectiveschedulingwhereoneoftheperformancemeasuresisrobustness).Littlethe-oreticalresearchhasbeendoneontheseissues.Thistopicmaybecomeanimportantareaofresearchinthenearfuture.15.3MachineLearningMechanismsInpractice,thealgorithmsembeddedinaplanningorschedulingsystemoftendonotyieldplansorschedulesthatareacceptabletotheuser.Theinade-quacyofthealgorithmsisbasedonthefactthatplanningandschedulingproblems(whichoftenhavemultipleobjectives)areinherentlyintractable.Itisextremelydifficulttodevelopalgorithmsthatcanprovideareasonableandacceptablesolutionforanyinstanceofaprobleminrealtime.
38015AdvancedConceptsinSystemsDesignNewresearchinitiativesarefocussingonthedesignanddevelopmentoflearningmechanismsthatenableplanningandschedulingsystemsthatareindailyusetoimprovetheirsolutiongenerationcapabilities.Thisprocessre-quiresasubstantialamountofexperimentalwork.Anumberofmachinelearn-ingmethodshavebeenstudiedwithregardtotheirapplicabilitytoplanningandscheduling.Thesemethodscanbecategorizedasfollows:(i)rotelearning,(ii)case-basedreasoning,(iii)inductionmethodsandneuralnetworks,(iv)classifiersystems.Thesefourclassesoflearningmechanismsareinwhatfollowsdescribedinsomemoredetail.Rotelearningisaformofbruteforcememorization.Thesystemsavesoldsolutionsthatgavegoodresultstogetherwiththeinstancesonwhichtheywereapplied.However,thereisnomechanismforgeneralizingthesesolutions.Thisformoflearningisonlyusefulwhenthenumberofpossibleplanningorschedulinginstancesislimited,i.e.,asmallnumberofjobsofveryfewdifferenttypes.Itisnotveryeffectiveinacomplexenvironment,whentheprobabilityofasimilarinstanceoccurringagainisverysmall.Case-basedreasoningattemptstoexploitexperiencegainedfromsimilarproblemssolvedinthepast.Aschedulingproblemrequirestheidentificationofsalientfeaturesofpastschedules,aninterpretationofthesefeatures,andamechanismfordeterminingwhichcasestoredinmemoryisthemostusefulinthecurrentcontext.Giventhelargenumberofinteractingconstraintsinherentinscheduling,existingcaseindexingschemesareofteninadequateforbuildingthecasebaseandsubsequentretrieval,andnewwayshavetobedeveloped.Thefollowingexampleshowshowtheperformanceofacompositedispatchingrulecanbeimprovedusingacrudeformofcase-basedreasoning;theformofcase-basedreasoningadoptedisoftenreferredtoastheparameteradjustmentmethod.Example15.3.1(Case-BasedReasoning:ParameterAdjustment).ConsiderasinglemachinewithnjobsandthetotalweightedtardinesswjTjastheobjectivetominimize.Moreover,thejobsaresubjecttosequencede-pendentsetuptimessjk.ThisproblemhasbeenconsideredinSection5.2andalsointhethirdsectionofAppendixC.Afairlyeffectivecompositedispatch-ingruleforthisschedulingproblemistheATCSrule.Whenthemachinehascompletedtheprocessingofjoblattimet,theATCSrulecalculatestherankingindexofjobjasIj(t,l)=wjpjexp−max(dj−pj−t,0)K1¯pexp−sljK2¯s,where¯sistheaveragesetuptimeofthejobsremainingtobescheduled,K1thescalingparameterforthefunctionoftheduedateofjobjandK2the
15.3MachineLearningMechanisms381scalingparameterforthesetuptimeofjobj.AsdescribedinAppendixC,thetwoscalingparametersK1andK2canberegardedasfunctionsofthreefactors:(i)theduedatetightnessfactorθ1,(ii)theduedaterangefactorθ2,(iii)thesetuptimeseverityfactorθ3=¯s/¯p.However,itisdifficulttofindappropriatefunctionsthatmapthethreefactorsintoappropriatevaluesforthescalingparametersK1andK2.Atthispointalearningmechanismmaybeuseful.Supposethatintheschedulingsystemtherearefunctionsthatmapcombinationsofthethreefactorsθ1,θ2andθ3ontotwovaluesforK1andK2.Thesedonothavetobealgebraicfunctions;theymaybetablesofdata.Whenaschedulinginstanceisconsidered,thesystemcomputesθ1,θ2andθ3andlooksinthecurrenttablesfortheappropriatevaluesofK1andK2.(ThesevaluesforK1andK2mayhavetobedeterminedbymeansofaninterpolation).TheinstanceisthensolvedusingthecompositedispatchingrulewiththesevaluesforK1andK2.Theobjectivevalueoftheschedulegeneratediscomputedaswell.However,inthatsamestep,withoutanyhumanintervention,thesystemalsosolvesthesameschedulinginstanceusingvaluesK1+δ,K1−δ,K2+δ,K2−δ(variouscombinations).Sincethedispatchingruleisveryfast,thiscanbedoneinrealtime.Theperformancemeasuresoftheschedulesgeneratedwiththeperturbedscalingparametersarealsocomputed.IfanyoftheseschedulesturnsouttobesubstantiallybetterthantheonegeneratedwiththeoriginalK1andK2,thentheremaybeareasonforchangingthemappingfromthecharacteristicfactorsontothescalingparameters.Thiscanbedoneinternallybythesystemwithoutanyinputfromtheuserofthesystem.Thelearningmechanismdescribedintheexampleaboveisanon-linemechanismthatoperateswithoutsupervision.Thismechanismisanexam-pleofcase-basedreasoningandcanbeappliedtomulti-objectiveplanningandschedulingproblemsaswell,evenwhenasimpleindexruledoesnotexist.Thethirdclassoflearningmechanismsareoftheinductiontype.Themostcommonformofaninductiontypelearningmechanismisaneuralnetwork.Aneuralnetconsistsofanumberofinterconnectedneuronsorunits.Theconnectionsbetweenunitshaveweights,whichrepresentthestrengthsoftheconnectionsbetweentheunits.Amulti-layerfeedforwardnetiscomposedofinputunits,hiddenunitsandoutputunits(seeFigure15.2).Aninputvectorisprocessedandpropagatedthroughthenetworkstartingattheinputunitsandproceedingthroughthehiddenunitsallthewaytotheoutputunits.Theactivationlevelofinputunitiissettotheithcomponentoftheinputvector.Thesevaluesarethenpropagatedtothehiddenunitsviatheweightedconnections.Theactivationlevelofeachhiddenunitisthencomputedbysummingtheseweightedvalues,andbytransformingthesum
38215AdvancedConceptsinSystemsDesignHidden unitsInput unitsOutputunitVi1Vi2Vi3Vi4OjFig.15.2.AFour-layerNeuralNetworkthroughafunctionf,thatis,al=fql,kwklak,wherealistheactivationlevelofunitl,qlisthebiasofunitl,andwklistheconnectionweightbetweennodeskandl.Theseactivationlevelsarepropa-gatedtotheoutputunitsviatheweightedconnectionsbetweenthehiddenunitsandtheoutputunitsandtransformedagainbymeansofthefunctionfabove.Theneuralnet’sresponsetoagiveninputvectoriscomposedoftheactivationlevelsoftheoutputunitsthatarereferredtoastheoutputvector.Thedimensionoftheoutputvectordoesnothavetobethesameasthedimensionoftheinputvector.Theknowledgeofthenetisstoredintheweightsandtherearewellknownmethodsforadjustingtheweightsontheconnectionsinordertoobtainap-propriateresponses.Foreachinputvectorthereisamostappropriateoutputvector.Alearningalgorithmcomputesthedifferencebetweentheoutputvec-torofthecurrentnetandthemostappropriateoutputvectorandsuggestsincrementaladjustmentstotheweights.Onesuchmethodiscalledtheback-propagationlearningalgorithm.Thenextexampleillustratestheapplicationofaneuralnettomachinescheduling.Example15.3.2(NeuralNetforParallelMachineScheduling).Con-sidermnon-identicalmachinesinparallel.Machineihasspeedviandifjobjisprocessedonmachinei,thenitsprocessingtimeispij=pj/vi.Thejobshavedifferentreleasedatesandduedatesandoneoftheobjectivesistominimizethetotalweightedtardiness(notethattheweightsintheobjective
15.3MachineLearningMechanisms383functionarenotrelatedtotheweightsintheneuralnet).Thejobsonama-chinearesubjecttosequencedependentsetuptimessjk.Foreachmachinethereisalreadyapartialscheduleinplacewhichconsistsofjobsalreadyas-signed;morejobsarereleasedastimegoeson.Ateachnewreleaseithastobedecidedtowhichmachinethejobshouldbeassigned.Theneuralnethastosupportthisdecision-makingprocess.Typically,theencodingofthedataintheformofinputvectorsiscrucialtotheproblemsolvingprocess.Intheparallelmachinesapplication,eachinputpatternrepresentsadescriptionoftheattributesofasequenceononeofthemachines.Thevaluesoftheattributesofasequencearedeterminedasfollows.Eachnewjobisfirstpositionedwhereitsprocessingtimeistheshortest(includingthesetuptimeimmediatelyprecedingitandthesetuptimeimmediatelyfollowingit).Afterthisinsertionthefollowingattributesarecomputedwithregardtothatmachine:(i)theincreaseinthetotalweightedcompletiontimesofallthejobsal-readyscheduledonthemachine;(ii)thenumberofadditionaljobsthatarelateonthemachine;(iii)theaverageadditionallatenessesofjobsalreadyscheduledonthema-chine;(iv)thecurrentnumberofjobsonthemachine.However,theimportanceofeachindividualattributeisrelative.Forex-ample,knowingthatthenumberofjobsonamachineisfivedoesnotmeanmuchwithoutknowingthenumberofjobsontheothermachines.LetNilbeattributelwithrespecttomachinei.TwotransformationshavetobeappliedtoNil.Translation:TheNilvalueofattributelunderagivensequenceonma-chineiistransformedasfollows.Nil=Nil−min(N1l,…,Nml)i=1,…,m,l=1,…,k.Inthisway,Ni∗l=0forthebestmachine,machinei∗,withrespecttoat-tributelandthevalueNilcorrespondstothedifferencewiththebestvalue.Normalization:TheNilvalueistransformedbynormalizingoverthemax-imumvalueinthecontext.Nil=Nilmax(N1l,…,Nml)i=1,…,m,l=1,…,k.Clearly,0≤Nil≤1.Thesetransformationsmakethecomparisonsoftheinputpatternscorre-spondingtothedifferentmachinessignificantlyeasier.Forexample,ifthevalueofattributelisimportantinthedecisionmakingprocess,thenama-chinewiththel-thattributeclosetozeroismorelikelytobeselected.AneuralnetarchitecturetodealwiththisproblemcanbeofthestructuredescribedinFigure15.2.Thisisathreelayernetworkwithfourinputnodes
38415AdvancedConceptsinSystemsDesign(equaltothenumberofattributesk),fourhiddennodesandoneoutputnode.Eachinputnodeisconnectedtoallthehiddennodesaswellastotheoutputnode.Thefourhiddennodesareconnectedtotheoutputnodeaswell.Duringthetrainingphaseofthisnetworkanextensivejobreleasepro-cesshastobeconsidered,say1000jobs.Eachjobreleasegeneratesminputpatterns(mbeingthenumberofmachines)whichhavetobefedintothenet-work.Duringthistrainingprocessthedesiredoutputofthenetissetequalto1whenthemachineassociatedwiththegiveninputisselectedbytheex-pertforthenewreleaseandequalto0otherwise.Foreachinputvectorthereisadesiredoutputandthelearningalgorithmhastocomputetheerrorordifferencebetweenthecurrentneuralnetoutputandthedesiredoutputinordertomakeincrementalchangesintheconnectionweights.Awell-knownlearningalgorithmfortheadjustmentofconnectionweightsisthebackprop-agationlearningalgorithm;thisalgorithmrequiresthechoiceofaso-calledlearningrateandamomentumterm.Attheconclusionofthetrainingphasetheconnectionweightsarefixed.Usingthenetworkafterthecompletionofthetrainingphaserequiresthateverytimeajobisreleasedtheminputpatternsarefedintothenet;themachineassociatedwiththeoutputclosestto1isthenselectedforthegivenjob.IncontrasttothelearningmechanismdescribedinExample15.3.1themechanismdescribedinExample15.3.2requiresoff-linelearningwithsuper-vision,i.e.,training.Thefourthclassoflearningmechanismsconsistsoftheclassifiersystems.Acommonformofclassifiersystemcanbeimplementedviaageneticalgorithm(seeAppendixC).However,achromosome(orstring)insuchaalgorithmnowdoesnotrepresentaschedule,butratheralistofrules(e.g.,priorityrules)thataretobeusedinthesuccessivesteps(oriterations)ofanalgorithmicframeworkdesignedtogenerateschedulesfortheproblemathand.Forex-ample,consideraframeworkforgeneratingjobshopschedulesthatissimilartoAlgorithmB.4.2.However,thisalgorithmcanbemodifiedbyreplacingitsStep3withapriorityrulethatselectsanoperationfromsetΩ.Ascheduleforthejobshopcannowbegeneratedbydoingnmsuccessiveiterationsofthisalgorithmwherenmisthetotalnumberofoperations.Everytimeanewoperationhastobescheduled,agivenpriorityruleisusedtoselecttheop-erationfromthecurrentsetΩ.Theinformationthatspecifieswhichpriorityruleshouldbeusedineachiterationcanbestoredinastringoflengthnmforageneticalgorithm.Thefitnessofsuchastringisthevalueoftheobjectivefunctionobtainedwhenallthe(local)rulesareappliedsuccessivelyinthegivenframework.Thisrepresentationofasolution(byspecifyingrules),how-ever,requiresrelativelyintricatecross-overoperatorsinordertogetfeasibleoff-spring.Thegeneticalgorithmisthususedtosearchoverthespaceofrulesandnotoverthespaceofactualschedules.Thegeneticalgorithmservesthiswayasameta-strategythatcontrolstheuseofpriorityrules.
15.4DesignofPlanningandSchedulingEnginesandAlgorithmLibraries38515.4DesignofPlanningandSchedulingEnginesandAlgorithmLibrariesAplanningorschedulingengineinasystemoftencontainsalibraryofal-gorithmicprocedures.Suchalibrarymayincludebasicdispatchingrules,compositedispatchingrules,shiftingbottlenecktechniques,localsearchtech-niques,branch-and-boundprocedures,beamsearchtechniques,mathematicalprogrammingroutines,andsoon.Foraspecificinstanceofaproblemoneproceduremaybemoresuitablethananother.Theappropriatenessofapro-ceduremaydependontheamountofCPUtimeavailableorthelengthoftimetheuseriswillingtowaitforasolution.Theuserofsuchaplanningorschedulingsystemmaywanttohaveacertainflexibilityintheusageofthevarioustypesofproceduresintheli-brary.Thedesiredflexibilitymaysimplyimplyanabilitytodeterminewhichproceduretoapplytothegiveninstanceoftheproblem,oritmayimplymoreelaboratewaysofmanipulatinganumberofprocedures.Aplanningorschedulingenginemayhavemodulesthatallowtheuser(i)toanalyzethedataofaninstanceanddeterminealgorithmicparame-ters,(ii)tosetupalgorithmsinparallel,(iii)tosetupalgorithmsinseries,(iv)tointegratealgorithms.Forexample,analgorithmlibraryallowsausertodothefollowing:hemayhavestatisticalproceduresathandwhichhecanapplytothedatasetinordertogeneratesomestatistics,suchasaverageprocessingtime,rangeofprocessingtimes,duedatetightness,setuptimeseverity,andsoon.Basedonthesestatisticstheusercanselectaprocedureandspecifyappropriatelevelsforitsparameters(e.g.,scalingparameters,lengthsoftabu-lists,beamwidths,numberofiterations,andsoon).Ifauserhasmorethanonecomputerorprocessorathisdisposal,hemaywanttoapplydifferentproceduresconcurrently(i.e.,inparallel),sincehemaynotknowinadvancewhichoneisthemostsuitablefortheinstanceundercon-sideration.Thedifferentproceduresfunctionthencompletelyindependentlyfromoneanother.Ausermayalsowanttoconcatenateproceduresi.e.,setvariousproceduresupinseries.Thatis,heorshewouldsettheproceduresupinsuchawaythattheoutputofoneservesasaninputtoanother,e.g.,theoutcomeofadispatchingruleservesastheinitialsolutionforalocalsearchprocedure.Thetransferofdatafromoneproceduretothenextisusuallyrelativelysimple.Forexample,itmaybejustaschedule,which,inthecaseofasinglemachine,isapermutationofthejobs.Inaparallelmachineenvironmentorajobshopenvironmentitmaybeacollectionofsequences,oneforeachmachine.Example15.4.1(ConcatenationofProcedures).Consideraschedulingenginethatallowsausertofeedtheoutcomeofacompositedispatching
38615AdvancedConceptsinSystemsDesignruleintoalocalsearchprocedure.Thismeansthattheoutputofthefirststage,i.e.,thedispatchingrule,isacompleteschedule.Thescheduleisfeasibleandthestartingtimesandcompletiontimesofalltheoperationshavebeendetermined.Theoutputdataofthisprocedure(andtheinputdataforthenextprocedure)maycontainthefollowinginformation:(i)thesequenceofoperationsoneachmachine;(ii)thestarttimeandcompletiontimeofeachoperation;(iii)thevaluesofspecificobjectivefunctions.Theoutputdatadoesnothavetocontainallthedatalistedabove;forex-ample,itmayincludeonlythesequenceofoperationsoneachmachine.Thesecondprocedure,i.e.,thelocalsearchprocedure,mayhavearoutinethatcancomputethestartandcompletiontimesofalltheoperationsgiventhestructureoftheproblemandthesequencesoftheoperations.Anotherlevelofflexibilityallowstheusernotonlytosetupproceduresinparallelorinseries,buttointegratetheproceduresinamorecomplexmanner.Whendifferentproceduresareintegratedwithinoneframework,theydonotworkindependentlyfromoneanother;theeffectivenessofoneproceduremaydependontheinputorfeedbackreceivedfromanother.Considerabranch-and-boundprocedureorabeamsearchprocedureforaschedulingproblem.Ateachnodeinthesearchtree,onehastoobtaineitheralowerboundoranestimateforthetotalpenaltythatwillbeincurredbythejobsthathavenotyetbeenscheduled.Alowerboundcanoftenbeobtainedbyassumingthattheversionofaproblemisofteneasiertosolvethanitsnonpreemptivecounterpartandtheoptimalsolutionofthepreemptiveproblemprovidesalowerboundfortheoptimalsolutionofthenonpreemptiveversion.Anotherexampleofanintegrationofproceduresarisesindecompositiontechniques.Amachine-baseddecompositionprocedureistypicallyaheuristicdesignedforacomplicatedschedulingproblemwithmanysubproblems.AframeworkforaprocedurethatisapplicabletothemainproblemcanbeconstructedasinChapter5.However,theusermaywanttobeabletospecify,knowingtheparticularproblemorinstance,whichproceduretoapplyonthesubproblem.Ifprocedureshavetobeintegrated,thenoneoftenhastoworkwithinageneralframework(sometimesalsoreferredtoasacontrolstructure)inwhichoneormorespecifictypesofsubproblemshavetobesolvedmanytimes.Theusermaywanttohavetheabilitytospecifycertainparameterswithinthisframework.Forexample,iftheframeworkisasearchtreeforabeamsearch,thentheuserwouldliketobeabletospecifythebeamwidthaswellasthefilterwidth.Thesubproblemthathastobesolvedateachnodeofthesearchtreehastoyield(withlittlecomputationaleffort)agoodestimateforthecontributiontotheobjectivebythosejobsthathavenotyetbeenscheduled.Thetransferofdatabetweenprocedureswithinanintegratedframeworkmaybecomplicated.Itmaybethecasethatdataconcerningasubsetofremainingjobscanbescheduledwhileallowingpreemptions.Apreemptive
15.4DesignofPlanningandSchedulingEnginesandAlgorithmLibraries387jobsorasubsetofoperationshastobetransferred.Itmayalsobethecasethatthemachinesarenotavailableatalltimes.Thepositionsofthejobsalreadyscheduledonthevariousmachinesmaybefixed,implyingthattheprocedurethathastoscheduletheremainingjobsmustknowatwhattimesthemachinesarestillavailable.Iftherearesequencedependentsetuptimes,thentheprocedurealsohastoknowwhichjobwasthelastoneprocessedoneachmachine,inordertocomputethesequencedependentsetuptimeforthenextjob.Example15.4.2(IntegrationofProceduresinaBranchingScheme).Considerabranch-and-boundapproachforasinglemachineproblemwiththetotalweightedtardinessobjectiveandjobsthataresubjecttosequencedependentsetuptimes.Moreover,themachinemaynotbeavailableforcertainperiodsoftimeduetomaintenance.Thejobsarescheduledinaforwardmanner,i.e.,apartialscheduleconsistsofasequenceofjobsthatstartsattimezero.Ateachnodeofthebranchingtreeaboundhastobeestablishedforthetotalweightedtardinessofthejobsstilltobescheduled.Ifaprocedureiscalledtogeneratealowerboundforallschedulesthataredescendantsfromanyparticularnode,thenthefollowinginputdatahastobeprovided:(i)thesetofjobsalreadyscheduledandthesetofjobsstilltobescheduled;(ii)thetimeperiodsthatthemachineremainsavailable;(iii)thelastjobinthecurrentpartialschedule(inordertodeterminethesequencedependentsetuptime).Theoutputdataoftheproceduremaycontainasequenceofoperationsaswellasalowerbound.Therequiredoutputmaybejustthelowerbound;theactualsequencemaynotbeofinterest.Iftherearenosetuptimesthenaschedulecanalsobegeneratedinabackwardmanner(sincethevalueofthemakespanisthenknowninadvance).Example15.4.3(IntegrationofProceduresinaDecompositionScheme).Considerashiftingbottleneckframeworkforaflexiblejobshopwithateachworkcenteranumberofmachinesinparallel.Ateachiterationasubsetoftheworkcentershasalreadybeenscheduledandanadditionalworkcentermustbescheduled.Thesequencesoftheoper-ationsattheworkcentersalreadyscheduledimplythattheoperationsoftheworkcentertobescheduledinthesubproblemissubjecttodelayedprece-denceconstraints.Whentheprocedureforthesubproblemiscalled,acertainamountofdatahastobetransferred.Thesedatamayinclude:(i)thereleasedateandduedateofeachoperation;(ii)theprecedenceconstraintsbetweenthevariousoperations;(iii)thenecessarydelaysthatgoalongwiththeprecedenceconstraints.Theoutputdataconsistsofthesequenceoftheoperationsaswellastheirstarttimesandcompletiontimes.Italsocontainsthevaluesofgivenperformancemeasures.
38815AdvancedConceptsinSystemsDesignItisclearthatthetypeofinformationandthestructureoftheinformationismorecomplicatedthaninasimpleconcatenationofprocedures.Theseformsofintegrationofprocedureshaveledtothedevelopmentofso-calleddescriptionlanguagesforplanningandscheduling.Adescriptionlan-guageisahighlevellanguagethatenablesaplanneroraschedulertowritethecodeforacomplexintegratedalgorithmusingonlyalimitednumberofconcisestatementsorcommands.Eachstatementinadescriptionlanguageinvolvestheapplicationofarelativelypowerfulprocedure.Forexample,astatementmaycarrytheinstructiontoapplyatabu-searchproceduretoagivensetofjobsinagivenmachineenvironment.Theinputtosuchastate-mentconsistsofthesetofjobs,themachineenvironment,theprocessingrestrictionsandconstraints,thelengthofthetabu-list,aninitialschedule,andthetotalnumberofiterations.Theoutputconsistsofthebestschedulegeneratedbythetabu-searchprocedure.Otherstatementscanbeusedforsettingupproceduresinparallelorconcatenateprocedures.15.5ReconfigurableSystemsThelasttwodecadeshavewitnessedthedevelopmentofalargenumberofplanningandschedulingsystemsinindustryandinacademia.Someofthesesystemsareapplication-specific,othersaregeneric.Inimplementationsapplication-specificsystemstendtodosomewhatbetterthangenericsystemsthatarecustomized.However,application-specificsystemsareoftenhardtomodifyandadapttochangingenvironments.Genericsystemsareusuallysomewhatbetterdesignedandmoremodular.Nevertheless,anycustomiza-tionofsuchsystemstypicallyrequiresasignificantinvestment.Consideringtheexperienceofthelasttwodecades,itappearsusefultoprovideguidelinesthatfacilitateandstandardizethedesignandthedevelop-mentofplanningandschedulingsystems.Effortshavetobemadetoprovideguidelinesaswellastoolsforsystemsdevelopment.Themostrecentdesignstendtobeobject-oriented.Therearemanyadvantagesinfollowinganobject-orienteddesignap-proachforthedevelopmentofaplanningorschedulingsystem.First,thedesignismodular,whichmakesmaintenanceandmodificationofthesys-temrelativelyeasy.Second,largesegmentsofthecodearereusable.Thisimpliesthattwosystemsthatareinherentlydifferentstillmayshareasignif-icantamountofcode.Third,thedesignerthinksintermsofthebehaviorofobjects,notinlowerleveldetail.Inotherwords,theobject-orienteddesignapproachcanspeedupthedesignprocessandseparatethedesignprocessfromitsimplementation.Objectorientedsystemsareusuallydesignedaroundtwobasicentities,namelyobjectsandmethods.Objectsrefertovarioustypesofentitiesorcon-cepts.Themostobviousonesarejobsandmachinesoractivitiesandresources.
15.5ReconfigurableSystems389However,aplanorascheduleisalsoanobjectandsoareuser-interfacecompo-nents,suchasbuttons,menusandcanvases.Therearetwobasicrelationshipsbetweenobjecttypes,namelytheis-arelationshipandthehas-arelationship.Accordingtoanis-arelationshiponeobjecttypeisaspecialcaseofanotherobjecttype.Accordingtoahas-arelationshipanobjecttypemayconsistofseveralotherobjecttypes.Objectsusuallycarryalongstaticinformation,re-ferredtoasattributes,anddynamicinformation,referredtoasthestateoftheobject.Anobjectmayhaveseveralattributesthataredescriptorsassoci-atedwiththeobject.Anobjectmaybeinanyoneofanumberofstates.Forexample,amachinemaybebusy,idle,orbrokendown.Achangeinthestateofanobjectisreferredtoasanevent.Amethodisimplementedinasystemthroughoneormoreoperators.Operatorsareusedtomanipulatetheattributescorrespondingtoobjectsandmayresultinchangesofobjectstates,i.e.,events.Ontheotherhand,eventsmaytriggeroperatorsaswell.Thesequenceofstatesofthedifferentobjectscanbedescribedbyastate-transitionoreventdiagram.Suchaneventdiagrammayrepresentthelinksbetweenoperatorsandevents.Anoperatormayberegardedasthemannerinwhichamethodisimplementedinthesoftware.Anygivenoperatormaybepartofseveralmethods.Somemethodsmaybeverybasicandcanbeusedforsimplemanipulationsofobjects,e.g.,apairwiseinterchangeoftwojobsinaschedule.Othersmaybeverysophisticated,suchasanintricateheuristicthatcanbeappliedtoagivensetofjobs(objects)inagivenmachineenvironment(alsoobjects).Theapplicationofamethodtoanobjectusuallytriggersanevent.Theapplicationofamethodtoanobjectmaycauseinformationtobetransmittedfromoneobjecttoanother.Suchatransmissionofinformationisusuallyreferredtoasamessage.Messagesrepresentinformation(orcontent)thataretransmittedfromoneobject(forexample,aschedule)viaamethodtoanotherobject(forexample,auserinterfacedisplay).Amessagemayconsistofsimpleattributesorofanentireobject.Messagesaretransmittedwheneventsoccur(causedbytheapplicationofmethodstoobjects).Messageshavebeenreferredtointheliteraturealsoasmemos.Thetransmissionofmessagesfromoneobjecttoanothercanbedescribedbyatransitioneventdiagram,andrequiresthespecificationofprotocols.Aplanningorschedulingsystemmaybeobject-orientedinitsconceptualdesignand/orinitsdevelopment.Asystemisobject-orientedinitsconceptualdesignifthedesignofthesystemisobject-orientedthroughout.Thisimpliesthateveryconceptusedandeveryfunctionalityofthesystemiseitheranob-jectoramethodofanobject(whetheritisinthedataorknowledgebase,thealgorithmlibrary,theplanningorschedulingengineortheuserinterfaces).Eventhelargestmoduleswithinthesystemareobjects,includingthealgo-rithmlibraryandtheuserinterfacemodules.Asystemisobject-orientedinitsdevelopmentifonlythemoredetaileddesignaspectsareobject-orientedandthecodeisbasedonaprogramminglanguagewithobject-orientedextensionssuchasC++.
39015AdvancedConceptsinSystemsDesignManyplanningandschedulingsystemsdevelopedinthepasthaveobject-orientedaspectsandtendtobeobject-orientedintheirdevelopment.Anum-berofthesesystemsalsohaveconceptualdesignaspectsthatareobject-oriented.Somerelyoninferenceenginesforthegenerationoffeasibleplansorschedulesandothersareconstraintbasedrelyingonconstraintpropaga-tionalgorithmsandsearch.Thesesystemsusuallydonothaveenginesthatperformverysophisticatedoptimization.Notthatmanysystemsimplementedinthepasthavebeendesignedac-cordingtoanobject-orientedphilosophythroughout.Someaspectsthataretypicallynotobject-orientedare:(i)thedesignofplanningandschedulingengines,(ii)thedesignoftheuserinterfacesand(iii)thespecificationoftheprecedence,routingandlayoutconstraints.Fewexistingengineshaveextensivelibrariesofalgorithmsattheirdisposalwhichareeasilyreconfigurableandthatwouldbenefitfromamodularobject-orienteddesign(anobject-orienteddesignwouldrequireadetailedspecifica-tionofoperatorsandmethods).Sincemostplanningandschedulingenviron-mentswouldbenefitfromhighlyinteractiveoptimization,schedulegeneratorshavetobestronglylinkedtointerfacesthatallowschedulerstomanipulateschedulesmanually.Still,object-orienteddesignhasnothadyetamajorim-pactonthedesignofuserinterfacesforschedulingsystems.Theprecedenceconstraints,theroutingconstraints,andthemachinelayoutconstraintsareoftenrepresentedbyrulesinaknowledgebaseandaninferenceenginemustgenerateaplanorschedulethatsatisfiestherules.However,theseconstraintscanbemodeledconceptuallyeasilyusinggraphandtreeobjectsthatthencanbeusedbyanobjectorientedplanningorschedulingengine.15.6Web-BasedPlanningandSchedulingSystemsWiththeongoingdevelopmentininformationtechnology,conventionalsingle-userstand-alonesystemshavebecomeavailableinnetworksandontheInter-net.Basicallytherearethreetypesofweb-basedsystems:(i)informationaccesssystems,(ii)informationcoordinationsystems,(iii)informationprocessingsystems.Ininformationaccesssystems,informationcanberetrievedandsharedthroughtheInternet,throughEDIorthroughotherelectronicsystems.Theserveractsasaninformationrepositoryanddistributioncenter,suchasahomepageontheInternet.Ininformationcoordinationsystems,informationcanbegeneratedaswellasretrievedbymanyusers(clients).Theinformationflowsgoinmanydirec-tionsandtheservercansynchronizeandmanagetheinformation,suchasinprojectmanagementandinelectronicmarkets.
15.6Web-BasedPlanningandSchedulingSystems391Ininformationprocessingsystemstheserverscanprocesstheinformationandreturntheresultsofthisprocessingtotheclients.Inthiscase,theserversfunctionasapplicationprogramsthataretransparenttotheusers.Web-basedplanningandschedulingsystemsareinformationprocessingsystemsthatareverysimilartotheinteractiveplanningorschedulingsystemsdescribedinprevioussections,exceptthataweb-basedplanningorschedulingsystemisusuallyastronglydistributedsystem.Becauseoftheclient-serverarchitectureoftheInternet,alltheimportantcomponentsofaplanningorschedulingsystem,i.e.,itsdatabase,itsengine,anditsuserinterface,mayhavetobeadapted.Theremainingpartofthissectionfocusesonsomeofthetypicaldesignfeaturesofweb-basedplanningandschedulingsystems.Theadvantagesofhavingserversthatmakeplanningandschedulingsys-temsavailableonthewebarethefollowing.First,theinput-outputinterfaces(usedforthegraphicaldisplays)canbesupportedbylocalhostsratherthanbyserversatremotesites.Second,theserveraswellasthelocalclientscanhandlethedatastorageandmanipulation.Thismayalleviatetheworkloadattheserversitesandgivelocalusersthecapabilityandflexibilitytomanagethedatabase.Third,multipleserverscancollaborateonthesolutionoflarge-scaleandcomplicatedplanningandschedulingproblems.Asingleservercanpro-videapartialsolutionandtheentireproblemcanbesolvedusingdistributedcomputationalresources.Inordertoretainallthefunctionsinherentinaninteractiveplanningorschedulingsystem,themaincomponentsofasystemhavetoberestructuredinordertocomplywiththeclient-serverarchitectureandtoachievethead-vantageslistedabove.Thisrestructuringaffectsthedesignofthedatabase,theengineaswellastheuserinterface.Thedesignofthedatabasehasthefollowingcharacteristics:Theprocessmanageraswellastheplanningorschedulingmanagerresideattheservers.However,somedatacanbekeptattheclientfordisplayorfurtherprocessing.BoththeGanttchartandthedispatchlistsarerepresentationsofthesolutiongeneratedbytheengine.Thelocalclientcancachetheresultsforfastdisplayandfurtherprocessing,suchasediting.Similarly,boththeserverandtheclientcanprocesstheinformation.Figure15.3exhibitstheinformationflowbetweentheserverandlocalclients.Aclientmayhaveageneralpurposedatabasemanagementsystem(suchasSybaseorExcel)oranapplication-specificplanningorschedulingdatabasefordatastorageandmanipulation.Thedesignoftheplanningorschedulingenginehasthefollowingchar-acteristics:Alocalclientcanselectfortheproblemthathehastodealwithanalgorithmfromalibrarythatresidesataremoteserver.Often,thereisnoalgorithmspecificallydesignedforhisparticularplanningorschedulingproblemandhemaywanttocreateacompositeprocedureusingsomeofthealgorithmsthatareavailableinthelibrary.Theserverorclientalgorithmgeneratormayfunctionasaworkplaceforuserstocreatenewprocedures.Figure15.4showshowanewcompositeprocedurecanresultinanewalgo-rithmthatthencanbeincludedinboththeserverandtheclientlibraries.
39215AdvancedConceptsinSystemsDesignProcess informationSchedule informationLocal databaseServer databaseClientUser interfaceServerScheduling engineInternetFig.15.3.InformationFlowBetweenServerandClientScheduler dispatcherInternetAlgorithm generatorLocal algorithm libraryServer algorithm libraryServerClientChoose algorithmAdd algorithmFig.15.4.ProcessofConstructingNewMethods
15.7Discussion393Thislocalworkplacecanspeeduptheprocessofconstructingintermediateandfinalcompositemethodsandextendtheserverandclientlibrariesatthesametime.TheInternetalsohasaneffectonthedesignoftheuserinterfaces.Us-ingexistingInternetsupport,suchasHTML(HyperTextMarkupLanguage),Java,Javascript,PerlandCGI(CommonGatewayInterface)functions,thegraphicaluserinterfacesofplanningandschedulingsystemscanbeimple-mentedaslibraryfunctionsattheserversites.Throughtheuseofappropriatebrowsers(suchasNetscape),userscanenterdataorviewscheduleswithadynamichypertextinterface.Moreover,ausercanalsodevelopinterfacedis-playsthatlinkserverinterfacefunctionstootherapplications.InFigure15.3itisshownhowdisplayfunctionscanbesupportedeitherbyremoteserversorbylocalclients.Thusitisclearthatserverscanbedesignedinsuchawaythattheycanhelplocalclientssolvetheirplanningorschedulingproblems.Thelocalclientscanmanipulatedataandconstructnewplanningandschedulingmethods.Theserversfunctionasregularinteractiveplanningandschedulingsystemsexceptthatnowtheycanbeusedinamulti-userenvironmentontheInternet.Web-basedplanningandschedulingsystemscanbeusedinseveralways.Onewayisbasedonpersonalcustomizationandanotherwayisbasedonunionization.Personalcustomizationimpliesthatasystemcanbecustomizedtosatisfyanindividualuser’sneeds.Differentusersmayhavedifferentre-quirements,sinceeachhashisownwayofusinginformation,applyingplan-ningandschedulingprocedures,andsolvingproblems.Personalizedsystemscanprovideshortcutsandimprovesystemperformance.Unionizationmeansthataweb-basedplanningorschedulingsystemcanbeusedinadistributedenvironment.Adistributedsystemcanexchangeinformationefficientlyandcollaborateeffectivelyinsolvinghardplanningandschedulingproblems.WiththedevelopmentofInternettechnologyandclient-serverarchitec-tures,newtoolscanbeincorporatedinplanningandschedulingsystemsforsolvinglarge-scaleandcomplicatedproblems.Itappearsthatweb-basedsys-temsmaywellleadtoviablepersonalizedinteractiveplanningandschedulingsystems.15.7DiscussionManyteamsinindustryandacademiaarecurrentlydevelopingplanningandschedulingsystems.Thedatabase(orobjectbase)managementsystemsareusuallyoff-the-shelf,developedbycompaniesthatspecializeinthesesystems,e.g.,Oracle.Thesecommercialdatabasesaretypicallynotspecificallygearedforplanningandschedulingapplications;theyareofamoregenericnature.Dozensofsoftwaredevelopmentandconsultingcompaniesspecializeinplanningandschedulingapplications.Theymayspecializeevenincertainniches,e.g.,planningandschedulingapplicationsintheprocessindustries
39415AdvancedConceptsinSystemsDesignorinthemicroelectronicsindustries.Eachofthesecompanieshasitsownsystemswithelaborateuserinterfacesanditsownwayofdoinginteractiveoptimization.Researchanddevelopmentinplanningandschedulingalgorithmsandinlearningmechanismswillmostlikelyonlytakeplaceinacademiaorinlargeindustrialresearchcenters.Thistypeofresearchneedsextensiveexperimen-tation;softwarehousesoftendonothavethetimeforsuchdevelopments.Inthefuture,theInternetmayallowforthefollowingtypesofinteractionbetweensoftwarecompaniesanduniversitiesthatdevelopsystemsontheonesideandcompaniesthatneedplanningandschedulingservices(customers)ontheotherside.Acustomermayuseasystemthatisavailableontheweb,enteritsdataandrunthesystem.Thesystemgivesthecustomerthevaluesoftheperformancemeasuresofthesolutiongenerated.However,thecustomercannotyetseetheplanorschedule.Iftheperformancemeasuresofthesolutionaretothelikingofthecustomer,thenhemaydecidetopurchasethesolutionfromthecompanythatownsthesystem.Exercises15.1.Onewayofconstructingrobustschedulesisbyinsertingidletimes.De-scribeallthefactorsthatinfluencethetiming,thefrequencyandthedurationoftheidleperiods.15.2.Considerallthenonpreemptiveschedulesonasinglemachinewithnjobs.Defineameasureforthe”distance”(orthe”difference”)betweentwoschedules.(a)Applythemeasurewhenthetwoschedulesconsistofthesamesetofjobs.(b)Applythemeasurewhenonesetofjobshasonemorejobthantheotherset.15.3.ConsiderthesamesetofjobsasinExample15.2.1.Assumethatthereisaprobabilitypthatthemachineneedsservicingbeginningattime2.Theservicingtakes10timeunits.(a)Assumethatneitherpreemptionnorresequencingisallowed(i.e.,aftertheservicinghasbeencompleted,themachinehastocontinueprocessingthejobitwasprocessingbeforetheservicing).Determinetheoptimalsequence(s)asafunctionofp.(b)Assumepreemptionisnotallowedbutresequencingisallowed.Thatis,afterthefirstjobhasbeencompletedtheschedulermaydecidenottostartthejobheoriginallyscheduledtogosecond.Determinetheoptimalsequence(s)asafunctionofp.(c)Assumepreemptionaswellasresequencingareallowed.Determinetheoptimalsequence(s)asafunctionofp.
Exercises39515.4.Considertwomachinesinparallelthatoperateatthesamespeedandtwojobs.Theprocessingtimesofeachoneofthetwojobsisequaltoonetimeunit.Ateachpointintimeeachmachinehasaprobability0.5ofbreakingdownforonetimeunit.Job1canonlybeprocessedonmachine1whereasjob2canbeprocessedoneitheroneofthetwomachines.ComputetheexpectedmakespanundertheLeastFlexibleJobfirst(LFJ)ruleandundertheMostFlexibleJobfirst(MFJ)rule.15.5.Considerasinglemachineschedulingproblemwiththejobsbeingsub-jecttosequencedependentsetuptimes.Defineameasureofjobflexibilitythatisbasedonthesetuptimestructure.15.6.Considerthefollowinginstanceofasinglemachinewithsequencede-pendentsetuptimes.Theobjectivetobeminimizedisthemakespan.Thereare6jobs.Thesequencedependentsetuptimesarespecifiedinthetablebelow.k0123456s0k-11+K1+1+Ks1kK-11+K1+1+s2k1+K-11+K1+s3k1+1+K-11+Ks4kK1+1+K-11+s5k1+K1+1+K-1s6k11+K1+1+K-AssumeKtobeverylarge.Defineastheneighbourhoodofascheduleallschedulesthatcanbeobtainedthroughanadjacentpairwiseinterchange.(a)Findtheoptimalsequence.(b)Determinethemakespansofallschedulesthatareneighborsoftheoptimalschedule.(c)Findaschedule,withamakespanlessthanK,ofwhichallneighborshavethesamemakespan.(Theoptimalsequencemaybedescribedasa“brit-tle”sequence,whilethelastsequencemaybedescribedasamore“robust”sequence.)15.7.ConsideraflowshopwithlimitedintermediatestoragesthatissubjecttoacyclicscheduleasdescribedinSection6.2.Machineinowhasatthecompletionofeachoperationaprobabilitypithatitgoesdownforanamountoftimexi.a)Defineameasureforthecongestionlevelofamachine.b)Supposethatoriginallytherearenobuffersbetweenmachines.Nowatotalofkbufferspacescanbeinsertedbetweenthemmachinesandtheallocationhastobedoneinsuchawaythattheschedulesareasrobustaspossible.Howdoestheallocationofthebufferspacesdependonthecongestionlevelsatthevariousmachines?
39615AdvancedConceptsinSystemsDesign15.8.Explainwhyrotelearningisanextremeformofcase-basedreasoning.15.9.Describehowabranch-and-boundapproachcanbeappliedtoaschedul-ingproblemwithmidenticalmachinesinparallel,thejobssubjecttosequencedependentsetuptimesandthetotalweightedtardinessasobjective.Thatis,generalizethediscussioninExample15.4.2toparallelmachines.15.10.ConsiderExample15.4.3andExercise15.9.Integratetheideaspre-sentedinanalgorithmfortheflexiblejobshopproblem.15.11.Consideraschedulingdescriptionlanguagethatincludesstatementsthatcancalldifferentschedulingproceduresforaschedulingproblemwithmidenticalmachinesinparallel,thetotalweightedtardinessobjectiveandthenjobsreleasedatdifferentpointsintime.Writethespecificationsfortheinputandtheoutputdataforthreestatementsthatcorrespondtothreeproceduresofyourchoice.Developalsoastatementforsettingtheproceduresupinparallelandastatementforsettingtheproceduresupinseries.Specifyforeachoneoftheselasttwostatementstheappropriateinputandoutputdata.15.12.Supposeaschedulingdescriptionlanguageisusedforcodingtheshift-ingbottleneckprocedure.Describethetypeofstatementsthatarerequiredforsuchacode.CommentsandReferencesThereisanextensiveliteratureonplanningandschedulingunderuncertainty(i.e.,PERT,stochasticscheduling).However,theliteratureonPERTandonstochasticscheduling,ingeneral,doesnotaddresstheissueofrobustnessperse.Butrobustnessconceptshavereceivedsomespecialattentionintheliterature;see,forexample,theworkbyLeonandWu(1994),Leon,WuandStorer(1994),MehtaandUzsoy(1999),Wu,StorerandChang(1991),Wu,ByeonandStorer(1999).Foranoverviewofresearchinreactiveplanningandscheduling,seetheexcellentsurveybySmith(1992)andtheframeworkpresentedbyVieira,HerrmannandLin(2003).Formoredetailedworkonreactiveschedulingandreschedulinginjobshops,seeBierwirthandMattfeld(1999)andSabuncuogluandBayiz(2000).Foranindustrialapplicationofreactivescheduling,seeElkamelandMohindra(1999).Researchonlearningmechanismsinplanningandschedulingsystemsstartedinthelateeighties;see,forexample,Shaw(1988),ShawandWhinston(1989),Yih(1990),Shaw,Park,andRaman(1992),andPark,RamanandShaw(1997).TheparametricadjustmentmethodfortheATCSruleinExample15.3.1isduetoChaoandPinedo(1992).AnexcellentoverviewoflearningmechanismsforschedulingsystemsispresentedinAytug,Bhattacharyya,KoehlerandSnowdon(1994).ThebookbyPesch(1994)focusesonlearninginschedulingthroughgeneticalgorithms(classifiersystems).Afairamountofdevelopmentworkhasbeendonerecentlyonthedesignofadaptableplanningandschedulingengines.Akkiraju,Keskinocak,MurthyandWu
CommentsandReferences397(1998,2001)discussthedesignofanagent-basedapproachforaschedulingsystemdevelopedatIBM.Feldman(1999)describesindetailhowalgorithmscanbelinkedandintegratedandWebster(2000)presentstwoframeworksforadaptableschedulingalgorithms.Thedesign,developmentandimplementationofmodularorreconfigurableplan-ningandschedulingsystemsisoftenbasedonobjectsandmethods.Forobjectsandmethods,seeBooch(1994),Martin(1993),andYourdon(1994).Formodulardesignwithregardtodatabasesandknowledgebases,see,forexample,Collinot,LePapeandPinoteau(1988),FoxandSmith(1984),Smith(1992),andSmith,Muscettola,Matthys,OwandPotvin(1990)).Foraninterestingdesignofaschedulingengine,seeSauer(1993).AsystemdesignproposedbySmithandLassila(1994)extendsthemodularphilosophyforplanningandschedulingsystemsfartherthananyprevioussystem.ThisisalsothecasewiththeapproachbyYen(1995)andPinedoandYen(1997).ThepaperbyYen(1997)containsthematerialonweb-basedplanningandschedulingsystemsthatispresentedinSection15.6.
Chapter16WhatLiesAhead?16.1Introduction……………………………39916.2PlanningandSchedulinginManufacturing…..40016.3PlanningandSchedulinginServices………..40116.4SolutionMethods………………………..40316.5SystemsDevelopment…………………….40516.6Discussion……………………………..40616.1IntroductionWithsomanydifferenttypesofapplications,itisnotsurprisingthatthereissuchagreatvarietyofplanningandschedulingmodels.Moreover,thenumer-oussolutionmethodsprovideahostofproceduresforthemyriadofproblems.Anygivenapplicationtypicallyrequiresitsowntypeofplanningandschedul-ingengineaswellascustomizeduserinterfaces.Theoverallarchitectureofasystemmaythereforebeveryapplication-specific.Thedecisionsupportsystemsthathavebeendesignedforplanningandschedulinginthevariousindustriestendtobequitedifferentfromoneanother.OverthelastdecadetherehasbeenatendencytobuildlargersystemsthathavemorecapabilitiesandthatarebetterintegratedwithintheERPsystemoftheenterprise.Especiallyinthemanufacturingworldtherehasbeenaten-dencytodesignanddevelopintegratedsystemswithmultiplefunctionalities.Especiallyinsupplychainmanagementthesystems(andtheirunderlyingmodels)havebecomemoreandmoreelaborate.Thedimensionsaccordingtowhichsuchasystemcanbemeasuredincludethenumberoffacilitiesinthenetworkaswellasthevarioustimehorizonsoverwhichthesystemmustoptimize.Integrationmayoccurinscope,spaceandtime.© Springer Science + Business Media, LLC 2009M.L. Pinedo, Planning and Scheduling in Manufacturing and Services,DOI: 10.1007/978-1-4419-0910-7_16,399
40016WhatLiesAhead?Inserviceorganizationsthesystemshavebecomemoreandmoreinte-gratedaswell.Inairlines,fleetschedulingsystemsandcrewschedulingsys-temshavebecomemoreandmoreintegratedaswellasinteractive.Incallcenters,personnelschedulingsystemsinteractwithoperatorassignmentandcallroutingsystems.Thedifficultiesinthemodeling,thedesign,andthedevelopmentoflargeintegratedsystemslietypicallyinthecoordinationandintegrationofsmallermoduleswithanarrowerscope.Therearemanyformsofintegrationandmanytypesofinterfacesbetweenmodules.Forexample,withinasystemamediumtermplanningmodulemayhavetointeractwithashorttermschedulingmodule;ashorttermschedulingmodulemayhavetointeractwithreactiveschedulingprocedures.Themodelsthathavebeendiscussedintheopenliter-atureandthatarecoveredinthisbooktendtobenarrowinscope;integrationaffectsthemodelingaswellasthesolutionmethodstobeused.Thedifficultiesoftentendtolieontheinterfacesbetweenthedifferentmodelingparadigms.Becauseoftheseformsofintegration,thevariousmodulesinasystem(thataredesignedtodealwithdifferenttypesofproblems)mustexchangedatawithoneanother.Sincethesolutionmethodinonemoduleoftenat-temptstoperformsomeformoflocaloptimizationinaniterativeman-ner,itmaybethecasethatthevariousmodulesmustexchangedatawithoneanotherregularlyinordertosolvetheirproblems.Becausethesolu-tionmethodsinthedifferentmodulesofasystemoftenattempttoop-timizeatdifferentlevels(withregardtothehorizon,thelevelofdetail,etc.),itmaybethecasethatinthetransferofdatabetweenthemod-ulesthedatahavetoundergosomeformoftransformation(e.g.,aggrega-tion).16.2PlanningandSchedulinginManufacturingManydifferentplanningandschedulingmodelsinmanufacturinghavebeenanalyzedintheliteratureindetail.PartIIofthisbookfocusesonsomeofthemoreimportantmodelcategories,namely(i)projectplanningandscheduling,(ii)machineschedulingandjobshopscheduling,(iii)schedulingofflexibleassemblysystems,(iv)economiclotscheduling,and(v)planningandschedulinginsupplychains.Clearly,PartIIdoesnotcoveralltheplanningandschedulingmodelsinmanufacturingthathavebeenconsideredintheliterature.Someofthemorenarrowareasinmanufacturingwithveryspecialschedulingproblemshavenotbeencoveredinthisbook.Examplesofsuchnicheareasare:(i)cranescheduling,and(ii)schedulingofroboticcells.
16.3PlanningandSchedulinginServices401Inboththeseareasafairlylargenumberofpapershaveappeared.However,thestructureoftheseproblemstendtobeveryspecialanditisnotclearofhowmuchusetheyaretootherplanningandschedulingareas.Someotherclassesofmodelshavereceivedlittleornoattentionintheliteratureandarethereforenotdiscussedinthisbookeither.Forexample,notmuchworkhasbeendoneonmodelsthatcombinecontinuousaspectsofschedulingproblemswithdiscreteaspects.IntheprocessindustriesitisoftenthecasethatplanningandschedulinginvolvesMake-To-Order(MTO)productionaswellasMake-To-Stock(MTS),i.e.,itmaybethecasethatacertainpartoftheproductionisMTOandtheremainingisMTS.TheMTOpartconsistsofasetoforderswithcommittedshippingdates(duedates),whereastheMTSpartisconcernedwithsetupcosts,inventorycarryingcosts,andstockoutprobabilities.TheMTOparthasstrongdiscreteaspectswhereastheMTSparthasstrongcontinuousaspects.Eventhoughthishybridtypeofschedulingproblemisverycommoninvariousprocessindustries,verylittleresearchhasbeendoneinthisarea.Suchmodelsmaygainmoreresearchattentioninthenearfuture.Anotherclassofmodelsthathasnotreceivedmuchattentionintheliter-aturearemodelsthatdealwiththeconceptofproductassemblyorproductionsynchronization,i.e.,partsandrawmaterialhavetoarriveatapointofas-semblyatmoreorlessthesametime,justbeforetheassemblyisabouttostart.Iftwoproductionprocesseshavetobesynchronizedinsuchawaythatthecompletionofoneiteminoneprocesshastooccurmoreorlessatthesametimeasthecompletionofthecorrespondingitemintheotherprocess,thentheassociatedschedulingproblemstendtobehard.Someclassesofmodelsalreadybeenconsideredintheliteraturemayinthefuturebegeneralizedandextendedinnewdirections.First,modelscon-sideredintheliteratureoftenfocusonasingleobjective.Inpractice,itmayverywellbethecasethatseveralobjectiveshavetobeconsideredatthesametimeandthattheuserwouldliketoseeaparametricanalysisinordertoevaluatethetrade-offs.Second,modelsthathavebeenanalyzedundertheassumptionthatthesolutionprocedurewouldbeusedinoneparticularman-ner(e.g.,allowingovernightnumbercrunching)mayhavetobestudiedagainundertheassumptionthattheprocedurewouldbeusedinadifferentmanner(e.g.,inrealtimeorinadistributedfashion).Third,modelswhichassumethataplanorschedulecanbecreatedentirelyfromscratch(withoutanyini-tialconditionsorwithoutapartialschedulealreadyinplace)mayhavetobegeneralizedtoallowfortheexistenceofinitialconditionsorapartialschedule.16.3PlanningandSchedulinginServicesAnumberofimportantapplicationareasinserviceshavebeencoveredinthisbook,namely(i)projectplanningandscheduling,
40216WhatLiesAhead?(ii)schedulingofmeetings,exams,andsoon,(iii)schedulingofentertainmentandsportevents,(iv)transportationscheduling,and(v)workforcescheduling.Allfiveareashavereceivedasignificantamountofattentionfromtheaca-demiccommunity,developers(softwarehouses),andusers.Thefiveareaslistedhaveverydifferentcharacteristics.Thefirst,second,andfifthareaaresomewhatgenericandareimportantinmanydifferentindustries;thethirdandfourthapplicationareaareveryindustry-specific.Systemsinthefirstandsecondapplicationareaseemtobeeasiertode-signandimplementthansystemsintheotherthreeareas.Projectplanning,schedulingofmeetingsandexams,aswellasrosteringandtimetablingtendtobesomewhatlessdifficulttodevelopanddonotrequireasignificantamountofinterfacingwithotherenterprise-widesystems.Thethirdapplicationarea,tournamentscheduling,typicallydoesnotin-volveanyrealtimescheduling.Tournamentschedulesarecreatedinadvanceandthereareusuallynoconstraintsonthecomputertime.Thealgorithmsmaybecomplicatedbutthesystemisusuallyastand-alonesystem.Systemsimplementationsinthefourthandfifthapplicationareatendtobemorechallenging.Intheseapplicationareasitisoftenthecasethatplanningandschedulingsystemshavetobelinkedtootherdecisionsupportsystems.Itmayinvolve,forexample,(i)integrationoffleetschedulingandcrewschedulingintransportation,(ii)integrationoffleetschedulingandyieldmanagementintransportation,and(iii)integrationofpersonnelschedulingandqueuemanagementincallcen-ters.Eventhoughthesystemshavebecomemoreandmoreintegrated,thedifferentproblemareasstilltendtobeanalyzedseparately.Thatis,notmuchworkhasbeendoneonhybridmodelsthatencompassdifferenttypesofproblemareasandhaveamoreglobalobjective.Justlikeinmanufacturing,therearevariousnicheareasinservicesthathavereceivedsomeresearchattentioninthepastandthathavenotbeenincludedinthisbook.Thesenicheareasinclude:(i)equipmentmaintenanceandorderscheduling;(ii)operatingroomschedulinginhospitals;(iii)schedulingofcheckprocessinginbanks.Inanequipmentmaintenancemodel,itistypicallyassumedthattherearemresourcesinparallel(e.g.,mdifferenttypesofrepairmen)andeachresourceiscapableofdoingacertaintypeoftask.Eachcustomerthatcomesinrequiresanumberofdifferentservices(e.g.,repairs),thatcanbedoneconcurrentlyandindependentlybythevariousresources.Acustomercandepartonlyafterallrequestedserviceshavebeencompleted.Eachcustomerisgiveninadvance
16.4SolutionMethods403atimewhenallhisservicesareexpectedtobecompleted;thistimeservesasaduedate.Oneobjectiveistominimizethetotalweightedtardiness.Thereareseveralapplicationsofplanningandschedulinginhealthcare,e.g.,nursescheduling(seeChapters12and13).Anotherimportantapplicationofplanningandschedulinginhealthcareconcernstheschedulingofoperatingroomsinhospitals.Schedulingofoperatingroomstendstobedifficultbecauseofitsstochasticnature.Thedurationsoftheoperationscanonlybeestimatedinadvance,buttheactualtimeshaveacertainvariability.Anadditionaldifficultyisthatasequenceofplannedsurgeriescanbeinterruptedattimesbyanemergencysurgery.Anobjectivefunctionmaybe,forexample,theminimizationoftheexpectedwaitingtimesofthepatients.Futureresearchinplanningandschedulinginservicesmayalsofocusongeneralizationsandextensionsofexistingmodels,similartotheextensionsmentionedwithregardtothemanufacturingmodels.Itmayinvolvemodelswithmultipleobjectives.Itmayinvolvemodelswhichassumethattheprob-lemshavetobesolvedinadifferentmode.Forexample,asystemmayhavetofunctioninareactivemode,i.e.,whensomethingunexpectedoccurs,thesys-temhastoreschedulealltheactivitiesinrealtime.Someresearchhasalreadybeendoneinthisdirection:airlinesrelyheavilyoncrewrecoveryprogramsincaseflightscheduleshavebeenthrownoutofwhackbecauseof,say,weatherconditions.16.4SolutionMethodsThedifferentsolutionmethods(describedinAppendixesA,B,C,andD)havevariousimportantcharacteristics.First,onecharacteristicisthequalityofthesolution;thatis,howclosetooptimalisthesolutiongenerated?Asecondcharacteristicisthecomputationtimeinvolved;thatis,doesthetech-niqueworkinarealtimeenvironmentordoesitneedasignificantamountofcomputertime?Athirdcharacteristicconcernsthedevelopmenttimeandeaseofmaintenance,e.g.,howeasyisittoadaptthecodetoaslightlydifferentproblem?Therearemanysolutionmethodsavailable,includingexactoptimizationtechniques(e.g.,integerprogramming),heuristics(e.g.,decompositiontech-niques),andconstraintprogrammingtechniques.Moreover,thesebasictech-niquescaninmanywaysbecombinedwithoneanotherintheformofhybridtechniques.Overthelastdecadeasignificantamountofresearchanddevelopmentworkhasbeendoneintheapplicationofintegerprogrammingtechniquestospecificschedulingproblems.Thetechniquesincludebranch-and-cut,branch-and-price(alsoreferredtoascolumngeneration),andbranch-cut-and-price.Themaingoalisalwaystoimprovetheperformanceoftheintegerprogram-mingtechniques;asignificantamountofprogresshasalreadybeenmade.
40416WhatLiesAhead?Anenormousamountofworkhasfocusedonheuristicmethods,includingdecompositiontechniquesandlocalsearchtechniques.Someofthisefforthasbeendirectedtorealtimeapplications;otherworkhasbeenbasedontheassumptionofanunlimitedamountofcomputertime.Duringthelastdecadeafairamountofefforthasfocusedonthedevel-opmentandimplementationofconstraintprogrammingapplications.Severalsoftwarefirms,includingILOGandFairIsaac(DashOptimization),havede-velopedelaboratesoftwarepackagesforconstraintprogrammingapplications.Theappropriatenessofatechniquedepends,ofcourse,alsoonthetypeofapplication(i.e.,thestructureoftheproblem)andthemannerinwhichthetechniqueissupposedtobeused(e.g.,inareactivemode).Bothmathe-maticalprogrammingandconstraintprogramminghavetheiradvantagesanddisadvantages;however,theirstrenghtsandweaknessesareoftencomplemen-tary.Ifthedatasetisfuzzyandsubjecttofrequentchanges,thenitmaynotmakesensetoimplementaveryexpensiveandtimeconsumingoptimizationtechnique.Insuchacaseaheuristicmaybemoreappropriate.Itisoftenadvantageoustodofirstananalysisconcerningtheaccuracyandrobustnessofthedataset.Thenextstageinthedevelopmentofsolutionmethodswillfocusonthedevelopmentofhybridtechniquesthatcombineconstraintprogrammingwithoptimizationapproachesandheuristics.Forexample,FairIsaac(DashOp-timization)recentlydevelopedtheXpress-CPconstraintprogrammingtool.ThisprogrammingtoolcanbeusedinconjunctionwiththeXpressoptimizertoolXpress-MPintheXpress-Mosellanguage.Xpress-CPcombinesthemath-ematicaloptimizationsoftwareXpress-MPandtheconstraintprogrammingsoftwareCHIPinahybridoptimizationframework.AnadvantageofXpress-CPisthatthemathematicalprogrammingandconstraintprogrammingmeth-odsareembeddedinthesamesoftwareenvironmentandtheproblemcanbeformulatedasonemodel.Suchanarchitecturefacilitatesthedevelopmentofamoresophisticatedsolutiontechniquethatusesmathematicalprogram-mingandcontraintprogrammingmethodologiesinanintegratedmanner.ThemodelcanbewrittenintheXpress-Mosellanguage;themathematicalpro-grammingandconstraintprogrammingsolverscanbeinvokedeasilywithoutaneedforintricateprogrammingoranyparticularexpertiseinthemethodolo-gies.BuiltonMosel’sNativeInterfacetechnology,theMoselmoduleXpress-CPprovidesahighlevelofabstractionthroughalargecollectionoftypesandconstraintstructuresthatsupportseveralfunctionsandprocedures.Inaddi-tion,theCPsolvercontainshighlevelprimitivestoguideduringthesearchprocedureandcanfollowvariousestablishedstrategiesandheuristics.This,togetherwithitsflexibledesign,makesXpress-CPausefultoolformodelingandsolvingcomplexschedulingproblems.Manyplanningandschedulingproblemshavemultipleobjectivesandtheweightsofthevariousobjectivesareeithersubjectiveorvaryovertime.Usersofplanningandschedulingsystemsmayattimesbeinterestedindoingsomeformofparametricanalysisanddeterminingthetrade-offs.Thisformofpara-
16.5SystemsDevelopment405metricoptimizationmayrequirespecialsolutiontechniquesaswell.Someofthesetechniquesmaybegeneric,whereasothersmaybeapplication-specific.Largeplanningandschedulingproblemsareoftendifficulttoanalyzeandmaybenefitfromdistributedprocessing.Ifaproblemishard,itmaybeadvan-tageoustodecompose(i.e.,partition)theproblemintosmallersubproblemswhichthenaresolvedseparatelyondifferentservers.Certaintypesofsolutiontechniquesmaybemoreamenabletodistributedprocessingthanothertypesofsolutiontechniques.Findingtheproperapproachestodecomposeproblemsandtoaggregatetheresultsofthevarioussubproblemsmayleadtochalleng-ingresearchinthefuture.Aninterestingexampleofsuchaframeworkistheso-calledA-TeamarchitectureofIBM,whichwasdevelopedatIBM’sT.J.WatsonResearchCenter.16.5SystemsDevelopmentAnenormousamountofsystemsdevelopmentisgoingoninmanufacturingaswellasinservices.Therearemanywaysinwhicheachmodulewithinasystemcanbedesigned(whetheritisthedatabase,theplanningorschedulingengine,ortheuserinterface).Inthedesignofasystem,thedatatransferandtheinformationexchangeplayanextremelyimportantrole.Supplychainmanagementneedsonagloballevelsystemsforlongterm,mediumterm,andshorttermoptimization;onalocallevelitneedssystemsforshorttermoptimizationaswell.Allthesesystemshavetoexchangedatawithoneanother.Theoutputofonesystem(e.g.,aplanorschedule)maybeaninputforanother.However,thedatausuallycannotbeexchangedwithoutsomeappropriatetransformation(eithersomeformofaggregationorsomeformofdisaggregation).Theoptimalformsofcommunicationbetweendecisionsupportsystemsrequireacertainamountofresearch.Datatransferandinformationexchangehavebeenundergoingmajortransformationsrecently.BecauseofthetechnologicaladvancesthathavetakenplaceoverthelastdecadewiththeemergenceoftheInternet,significantchangesaretakingplacewithregardtothedesignandtheimplementationofdecisionsupportsystemsinsupplychainmanagement.TheInternethasasignificantimpactonvariousaspectsofplanningandschedulinginsupplychains.Inmanyscenariosplanningandschedulingfunctionsarebecomingmoreweb-based.Itenhancesthelevelofcommunicationbetweenthevariousstagesaswellaswithsuppliersandcustomers;ithasthepotentialofreducingbull-whipeffects.Itallowsfor(i)moreaccurate,moreaccessible,andmoretimelyinformation;(ii)theestablishmentofe-HubsandVendorManagedInventories(VMI);(iii)auctionsandB2Bcommunicationfortheprocurementofcommodityitems.Havingmoreaccurateandmoretimelyinformationhasasignificantpos-itiveeffectonshorttermand/orreactivescheduling,butnotamajoreffect
40616WhatLiesAhead?onlongtermplanning.Animportantissueishowtoaggregatedataandde-terminetheformandthetimingofthetransferoftheinformationobtained.Informationwithregardtodisruptionsinthesupplyorinthedemandareconveyednowmorerapidlyupstreamaswellasdownstream.TheMaterialRequirementsPlanningsystemmaybeabletoreactfasterandmarketingmaybeabletoadjustpricesquicker.Inordertoimproveinventorymanagement,companieshavebeendesigninge-HubsfortheirVendorManagedInventory(VMI)systemswhichprovidevendorsaccesstocurrentinformationconcerninginventorylevelsofproductstheyhavetosupply.Ane-Hubhastheeffectthatreleasedates,whichareimportantforshorttermschedulingprocesses,areknownmoreaccurately.Sothereislessvariabilityandschedulesaremorereliable.WithmultiplesuppliersofitemsthataresomewhatcommoditytheInter-netprovidesacapabilityforauctionsandbidding(B2B).AssemblyoperationsinparticularcanmakeuseofB2B.B2Btendstobeapplicabletocommodityitemsofwhichqualityspecificationsareeasilyestablishedandforwhichtherearemultiplevendors.Inserviceindustriesmanyelaboratesystemshavebeendesignedandinte-grationhasbecomeanimportantaspectofsystemdesignaswell.Anumberofsoftwarehouseshavebeenconsideringhowtointegratepersonnelschedul-ingwithcallroutingmechanisms.Suchsystemshavealsobecomeweb-based.Appointmentsystemsingeneraltendtobeonline.Thedevelopmentofspeciallanguagesandtoolswillfacilitatethedevel-opmentandmaintenanceoflargeintegratedsystems.Forexample,theavail-abilityoftoolssuchasXpress-MPfromFairIsaac(DashOptimization)andtheRavelanguagefromJeppesenSystems(CarmenSystems)havehadaverypositiveeffectonsystemsdevelopment.Inthefuture,genericaswellasapplication-specificsystemswillfollowmoreandmoreanobject-orienteddesign.Atthesametime,theremaybenewdevelopmentsindescriptionlanguagesforplanningandscheduling.Suchdevelopmentswillleadtomoreflexibilityinthedesignaswellasintheuseofplanningandschedulingsystems.Itwillalsobeeasiertoreconfigureasystemwhentherearechangesintheenvironment.16.6DiscussionThemainfactorsthatgovernthedesignfeaturesofasystemincludetheen-vironmentinwhichthesystemhastobeinstalledandthemannerinwhichitissupposedtobeused.Forexample,themannerinwhichthesystemusestheInternetandtheamountofdistributedprocessingthatisbeingdone,areimportantfactorsintheselectionofthesolutionapproach(e.g.,mathemat-icalprogramming,constraintprogramming,etc.).Onceadecisionhasbeenmadewithregardtotheapproach,thenthemodelformulation,thesolution
CommentsandReferences407technique,andthesystemdesigncannotbeseparatedfromoneanother.Alltheseaspectsarecloselylinkedandhaveamajorinfluenceononeanother.Asignificantamountofresearchanddevelopmentisbeingdoneineverydirectionandatalllevels.Moreelaborateandsophisticatedmodelsarebeingformulatedandanalyzedandmoreeffective(hybrid)algorithmsarebeingdesignedandmoreefficientoptimizationmodulesarebeingdeveloped.Sincethereissuchagreatvarietyinplanningandschedulingproblems,systemreconfigurability(fromthemodelingpointofviewaswellasfromthesolutionapproachpointofview)isoftheutmostimportance.Suchrequirementsarestimulatingtheresearchinobjectorienteddesignaswellasindescriptionlanguagesforplanningandscheduling.CommentsandReferencesKjenstad(1998)studiedcoordinationissuesintheplanningandschedulingofsupplychains.Pinedo,SeshadriandShantikumar(1999)analyzedtheinteractionsbetweenpersonnelschedulingincallcentersandothercallcenterfunctions.Thereisanextensiveliteratureontheschedulingofroboticcells;see,forex-ample,thesurveypapersbyCrama,Kats,VandeKlundertandLevner(2000)andDawande,Geismar,SethiandSriskandarajah(2005).Forsomeinterestingpapersoncraneandhoistscheduling,seeDaganzo(1989),Yih(1994),GeandYih(1995),Armstrong,GuandLei(1996),Lee,LeiandPinedo(1997),andLiu,JiangandZhou(2002).SungandYoon(1998),Leung,LiandPinedo(2005,2006)havefocusedonequipmentrepairandorderscheduling.Thereisabroadliteratureonschedulinginhealthcare,inparticularonoperatingroomscheduling;see,forexample,BlakeandCarter(1997,2002),BlakeandDonald(2002),andCayirliandVeral(2004).Rachlin,Goodwin,Murthy,Akkiraju,Wu,KumaranandDas(2001)describeIBM’sA-Teamarchitecture.Murthy,Akkiraju,Goodwin,Keskinocak,Rachlin,Wu,Kumaran,Yeh,Fuhrer,Aggarwal,Sturzenbecker,JayaramanandDaigle(1999)dis-cussacooperativemulti-objectivedecisionsupportforthepaperindustry.Akkiraju,Keskinocak,MurthyandWu(2001)describeanagent-basedapproachtomulti-machinescheduling.Keskinocak,Goodwin,Wu,AkkirajuandMurthy(2001)andSadeh,HildumandKjenstad(2003)focusondecisionsupportsystemsformanaginganelectronicsupplychain.Yen(1997)consideredinteractiveschedulingagentsontheInternet.AsignificantamountofresearchiscurrentlybeingdoneonhowtheInternetcanbeusedtosolveverylargeoptimizationproblems;see,forexample,Fourer(1998)andFourerandGoux(2001).Yen(1995)considerstheuseofschedulingdiscriptionlanguagesinthedevelop-mentofschedulingsystems.
AppendicesAMathematicalProgramming:FormulationsandApplications………………………………….411BExactOptimizationMethods……………………..423CHeuristicMethods……………………………..441DConstraintProgrammingMethods…………………465ESelectedSchedulingSystems……………………..475FTheLekinSystemUser’sGuide…………………..479References………………………………………487Notation………………………………………..519SubjectIndex……………………………………523NameIndex……………………………………..529
AppendixAMathematicalProgramming:FormulationsandApplicationsA.1Introduction……………………………411A.2LinearProgrammingFormulations………….411A.3NonlinearProgrammingFormulations……….414A.4IntegerProgrammingFormulations…………416A.5SetPartitioning,SetCovering,andSetPacking.418A.6DisjunctiveProgrammingFormulations……..419A.1IntroductionInthisappendixwegiveanoverviewofthetypesofproblemsthatcanbeformulatedasmathematicalprograms.Theapplicationsdiscussedconcernonlyplanningandschedulingproblems.InordertounderstandtheexamplesthereadershouldbefamiliarwiththenotationandterminologiesintroducedinChapters2and3.Thisappendixisaimedatpeoplewhoarealreadyfamiliarwithelementaryoperationsresearchtechniques.Itmakesanattempttoputvariousnotionsandproblemdefinitionsinperspective.Relativelylittlewillbesaidaboutthestandardsolutiontechniquesforsolvingtheseproblems.A.2LinearProgrammingFormulationsThemostbasicmathematicalprogramistheLinearProgram(LP).AnLPreferstoanoptimizationprobleminwhichtheobjectiveandtheconstraintsarelinearinthevariablestobedetermined.AnLPcanbeexpressedasfollows:minimizec1x1+c2x2+···+cnxn
412AMathematicalProgramming:FormulationsandApplicationssubjecttoa11x1+a12x2+···+a1nxn≤b1a21x1+a22x2+···+a2nxn≤b2…am1x1+am2x2+···+amnxn≤bmxj≥0forj=1,…,n.Theobjectiveistheminimizationofcosts.Thec1,…,cnvectorisreferredtoasthecostvector.Thevariablesx1,…,xnhavetobedeterminedsothattheobjectivefunctionc1x1+···+cnxnisminimized.Thecolumnvectora1j,…,amjisreferredtoasactivityvectorj.Thevalueofthevariablexjreferstothelevelatwhichthisactivityjisperformed.Theb1,…,bmisreferredtoastheresourcevector.Thefactthatinlinearprogrammingndenotesthenumberofactivitiesandinschedulingtheorynreferstothenumberofjobsisamerecoincidence;thatinlinearprogrammingmdenotesthenumberofresourcesandinschedulingtheorymreferstothenumberofmachinesisacoincidenceaswell.Therepresentationabovecanbewritteninmatrixform:minimize¯c¯xsubjecttoA¯x≤¯b¯x≥0.ThereareseveralalgorithmsorclassesofalgorithmsforsolvinganLP.Thetwomostimportantonesare(i)thesimplexmethodsand(ii)theinteriorpointmethods.Althoughsimplexmethodsworkverywellinpractice,itisnotknownifthereisanyversionofthesimplexmethodthatsolvestheLPprobleminpolynomialtime.ThebestknownexampleofaninteriorpointmethodistheKarmarkaralgorithm,whichisknowntosolvetheLPprobleminpolynomialtime.Manytextbookscoverthesesubjectsingreatdetail.Aspecialcaseofthelinearprogramistheso-calledtransportationproblem.InthetransportationproblemthematrixAhasaspecialform.Thematrixhasmncolumnsandm+nrowsandtakestheformA=¯10···00¯1···0…………00···¯1II···I
A.2LinearProgrammingFormulations413wherethe¯1isarowvectorwithn1’sandtheIisann×nidentitymatrix.Allbuttwoentriesineachcolumn(activity)ofthisAmatrixarezero;thetwononzeroentriesareequalto1.Thismatrixisassociatedwiththefollowingproblem.Considerasituationinwhichitemshavetobeshippedfrommsourcestondestinations.Acolumn(activity)intheAmatrixrepresentsaroutefromagivensourcetoagivendestination.Thecostassociatedwiththiscolumn(activity)isthecostoftransportingoneitemfromthegivensourcetothegivendestination.Thefirstmentriesintheb1,…,bm+nvectorrepresentthesuppliesatthemsources,whilethelastnentriesoftheb1,…,bm+nvectorrepresentthedemandsatthendifferentdestinations.Usuallyitisassumedthatthesumofthedemandsequalsthesumofthesuppliesandtheproblemistotransportalltheitemsfromthesourcestothedemandpointsandminimizethetotalcostincurred.(Whenthesumofthesuppliesislessthanthesumofthedemandsthereisnofeasiblesolutionandwhenthesumofthesuppliesislargerthanthesumofthedemandsanartificialdestinationcanbecreatedwherethesurplusissenttoatzerocost).ThematrixAofthetransportationproblemisanexampleofamatrixwiththeso-calledtotalunimodularityproperty.Amatrixhasthetotaluni-modularitypropertyifthedeterminantofeverysquaresubmatrixwithinthematrixhasavalue−1,0or1.Itcanbeeasilyverifiedthatthisisthecasewiththematrixofthetransportationproblem.Thistotalunimodularitypropertyhasanimportantconsequence:ifthevaluesofthesuppliesanddemandsareallintegers,thenthereisanoptimalsolutionx1,…,xn,thatisavectorofintegersandthesimplexmethodwillfindsuchasolution.Thetransportationproblemisimportantinplanningandschedulingthe-oryforanumberofreasons.First,thereareplanningandschedulingproblemsthatcanbeformulatedastransportationproblems.Second,transportationproblemsareoftenusedtoobtainboundsinbranch-and-boundproceduresforNP-hardplanningandschedulingproblems.Thefollowingexamplefocusesonaschedulingproblemthatcanbefor-mulatedasatransportationproblem.ExampleA.2.1(ParallelMachinesandtheTransportationProb-lem).Considermmachinesinparallel.Thespeedofmachineiisvi.Therearenidenticaljobsthatallrequirethesameamountofprocessing,say1unit.Ifjobjisprocessedonmachinei,itsprocessingtimeis1/vi.Preemptionsarenotallowed.IfjobjiscompletedatCjapenaltyhj(Cj)isincurred.Letthevariablexijkbeequalto1ifjobjisscheduledasthekthjobonmachineiand0otherwise.Sothevariablexijkisassociatedwithanactivity.Thecostofoperatingthisactivityatunitleveliscijk=hj(Cj)=hj(k/vi).Assumethatthereareatotalofn×mpositions(amaximumofnjobscanbeassignedtoanymachine).Clearly,notallpositionswillbefilled.Thenjobsareequivalenttothensourcesinthetransportationproblemandthe
414AMathematicalProgramming:FormulationsandApplicationsn×mpositionsarethedestinations.TheproblemcanbeformulatedeasilyasanLP.minimizemi=1nj=1nk=1cijkxijksubjecttomi=1nk=1xijk=1forj=1,…,nnj=1xijk≤1fori=1,…,m,k=1,…,nxijk≥0fori=1,…,m,j=1,…,n,k=1,…,nThefirstsetofconstraintsensuresthatjobjisassignedtooneandonlyoneposition.Thesecondsetofconstraintsensuresthateachpositioni,khasatmostonejobassignedtoit.ActuallyfromtheLPformulationitisnotimmediatelyclearthattheoptimalvaluesofthevariablesxijkhavetobeeither0or1.FromtheconstraintsitmayappearatfirstsightthatanoptimalsolutionoftheLPformulationmayresultinxijkvaluesbetween0and1.Becauseofthetotalunimodularityproperty,theconstraintsdonotspecificallyhavetorequirethatthevariablesbeeither0or1.Aspecialcaseofthetransportationproblemistheassignmentproblem.Atransportationproblemisreferredtoasanassignmentproblemwhenn=m(thenumberofsourcesisequaltothenumberofdestinations)andateachsourcethereisasupplyofexactlyoneitemandateachdestinationthereisademandofexactlyoneitem.Theassignmentproblemisalsoimportantinschedulingtheory.Singlemachineproblemswiththenjobshavingidenticalprocessingtimesoftencanbeformulatedasassignmentproblems.ExampleA.2.2(SingleMachineandtheAssignmentProblem).Con-sideraspecialcaseoftheproblemdiscussedinExampleA.2.1,namelythesinglemachineversion.Lettheprocessingtimeofeachjobbeequalto1.Theobjectiveisagainhj(Cj).Sotherearenjobsandnpositions,andtheassignmentofjobjtopositionkhascosthj(k)associatedwithit.A.3NonlinearProgrammingFormulationsANonlinearProgram(NLP)isageneralizationofalinearprogramthatallowstheobjectivefunctionandtheconstraintstobenonlinearinx1,…,xn.Undercertainconvexityassumptionsontheobjectivefunctionandtheconstraintstherearenecessaryandsufficiencyconditionsforasolutiontobe
A.3NonlinearProgrammingFormulations415optimal.TheseconditionsareintheliteraturereferredtoastheKuhn-Tuckerconditions.Soiftheobjectivefunctionandconstraintssatisfytherequiredconvexityassumptions,thentheoptimalityofasolutioncanbeverifiedeasilyviatheseconditions.Thereareanumberofmethodsforsolvingnonlinearprogrammingprob-lems.Thesemethodstendtobedifferentfromthemethodsusedforlinearprogrammingproblems.(However,theymayuselinearprogrammingmethodsassubroutinesintheiroverallframework.)Themostcommonlyusedmethodsfornonlinearprogrammingare:(i)gradientmethodsand(ii)penaltyandbarrierfunctionmethods.InthisbooknonlinearprogramsappearinSections4.5and7.4(theallocationofresourcesinprojectschedulingwithnonlinearcostsandthelotschedul-ingproblemswitharbitraryschedules).Inbothapplicationstheobjectivefunctionsarenonlinearandtheconstraintslinear.TheproblemdescribedinSection4.5canbesolvedeitherbyagradientmethodorbyapenaltyfunctionmethod.TheprobleminSection7.4,whichhasasingleinequal-ityconstraint,canbesolvedinadifferentmanner.Iftheconstraintisnottight,thentheproblemisanunconstrainedproblemandiseasy.Ifthecon-straintistight,thentheproblemisanonlinearprogammingproblemwithanequalityconstraintandthistypeofproblemcanbesolvedrelativelyeasilyasfollows.Consideranonlinearprogrammingproblemwithmultipleequalitycon-straints.Suchaproblemcanbetransformedintoanunconstrainedproblemusingso-calledLagrangianmultipliers.Forexample,considerthenonlinearprogrammingproblemminimizeg(x1,…,xn)subjecttof1(x1,…,xn)=0…fm(x1,…,xn)=0Thisproblemcanbetransformedintothefollowingunconstrainedopti-mizationproblemwiththeobjectivefunctionminimizeg(x1,…,xn)+λ1f1(x1,…,xn)+···+λmfm(x1,…,xn)Iftheoriginalobjectivefunctiong(x1,…,xn)satisfiescertainconvexitycon-ditionsthentheoptimalsolutioncanbeobtainedbytakingthepartialderiva-tiveoftheunconstrainedproblemwithrespecttoxjandsetthatequaltozero.Thisyieldsnequationswithn+munknowns(x1,…,xn,λ1,…,λm).
416AMathematicalProgramming:FormulationsandApplicationsThesenequationstogetherwiththeoriginalsetofmconstraintsresultinasystemofn+mequationswithn+munknowns.ThistechniqueistheoneusedinSection7.4.A.4IntegerProgrammingFormulationsAnIntegerProgram(IP)isalinearprogramwiththeadditionalrequirementthatthevariablesx1,…,xnhavetobeintegers.Ifonlyasubsetofthevari-ablesarerequiredtobeintegerandtheremainingonesareallowedtobereal,theproblemisreferredtoasaMixedIntegerProgram(MIP).IncontrastwiththeLP,anefficient(polynomialtime)algorithmfortheIPorMIPdoesnotexist.Manyschedulingproblemscanbeformulatedasintegerprograms.Inthissectionwegivetwoexamplesofintegerprogrammingformulations.ThefirstexampledescribesanintegerprogrammingformulationforthenonpreemptivesinglemachineproblemwiththetotalweightedcompletiontimewjCjasobjective.EventhoughthisproblemisquiteeasyandcanbesolvedbythesimpleWeightedShortestProcessingTimefirst(WSPT)priorityrule(seeSection5.2andAppendixC),theproblemstillservesasausefulexample.Theformulationisagenericoneandcanbeusedforschedulingproblemswithmultiplemachinesaswell.ExampleA.4.1(ASingleMachineandIntegerProgramming).Con-siderthenonpreemptivesinglemachineschedulingproblemwiththejobssubjecttoprecedenceconstraintsandthetotalweightedcompletiontimeob-jective.Letxjkdenotea0−1decisionvariablewhichassumesthevalue1ifjobjprecedesjobkinthesequenceand0otherwise.Thevaluesxjjhavetobe0forallj.Thecompletiontimeofjobjisthenequaltonk=1pkxkj+pj.Theintegerprogrammingformulationoftheproblemwithoutprecedencecon-straintsthusbecomesminimizenj=1nk=1wjpkxkj+nj=1wjpjsubjecttoxkj+xjk=1forj,k=1,…,n,j=k,xkj+xlk+xjl≥1forj,k,l=1,…,n,j=k,j=l,k=l,xjk∈{0,1}forj,k=1,…,n,xjj=0forj=1,…,n.Wecanreplacethethirdsetofconstraintswithacombinationof(i)asetoflinearconstraintsthatrequireallxjtobenonnegative,(ii)asetoflinearconstraintsrequiringallxjtobelessthanorequalto1and(iii)asetof
A.4IntegerProgrammingFormulations417constraintsrequiringallxjtobeinteger.Constraintsrequiringcertainprece-dencesbetweenthejobscanbeaddedeasilybyspecifyingthecorrespondingxjkvalues.Often,thereismorethanoneintegerprogrammingformulationofthesameproblem.InExerciseA.1adifferentintegerprogrammingformulationhastobedevelopedforthesameproblem.InwhatfollowsanintegerprogrammingformulationispresentedforthejobshopproblemwiththemakespanobjectivedescribedinChapter5.ThisintegerprogrammingformulationissimilartotheonepresentedinChapter4fortheworkforceconstrainedprojectschedulingproblem.ExampleA.4.2(AJobShopandIntegerProgramming).ConsiderajobshopasdescribedinChapter5.Therearenjobsandmmachines.Jobjhastobeprocessedonmachineiforadurationofpij.Therouteofeachoneofthejobsisgivenandrecirculationisallowed.Ifoperation(i,j)ofjobjonmachineihastobecompletedbeforeoperation(h,j)onmachinehisallowedtostart,thenthisroutingconstraintmayberegardedasaprecedenceconstraint(i,j)→(h,j).LetthearcsetAdenotethesetofallsuchrouting(precedence)constraints.Letxijtdenotea0−1decisionvariablethatisequalto1ifoperation(i,j)iscompletedexactlyattimetand0otherwise.LetHdenoteanupperboundforthemakespanCmax;suchanupperboundcanbeobtainedfairlyeasily(take,forexample,thesumofalltheprocessingtimesofalljobsoverallmachines).Sothecompletiontimeofoperation(i,j)canbeexpressedasCij=Ht=1txijt.Inordertoformulatetheintegerprogram,assumethatthemakespanCmaxisalsoadecisionvariable.(However,themakespandecisionvariableisclearlynota0−1variable.)Theintegerprogramthatminimizesthemakespancannowbeformulatedasfollows.minimizeCmaxsubjecttoHt=1txijt−Cmax≤0fori=1,…,m;j=1,…,nHt=1txijt+pij−Ht=1txhjt≤0for(i,j)→(h,j)∈A.xijt∈{0,1}fori=1,…,m;j=1,…,n;t=1,…,H.InSectionA.6ofthisappendixaso-calleddisjunctiveprogrammingfor-mulationisgivenforthisproblem.ThereareseveralmethodsforsolvingIntegerPrograms.ThesemethodsarediscussedinAppendixB.
418AMathematicalProgramming:FormulationsandApplicationsA.5SetPartitioning,SetCovering,andSetPackingThereareseveralspecialtypesofIntegerProgrammingformulationsthathavenumerousplanningandschedulingapplicationsinpractice.Thissectionfocusesonthreesuchformulations,namelySetPartitioning,SetCovering,andSetPacking.TheintegerprogrammingformulationoftheSetPartitioningproblemhasthefollowingstructure.minimizec1x1+c2x2+···+cnxnsubjecttoa11x1+a12x2+···+a1nxn=1a21x1+a22x2+···+a2nxn=1…am1x1+am2x2+···+amnxn=1xj∈{0,1}forj=1,…,n.Allaijvaluesareeither0or1.Whentheequalsigns(=)intheconstraintsarereplacedbygreaterthanorequal(≥),theproblemisreferredtoastheSetCoveringproblem,andwhentheequalsignsarereplacedbylessthanorequal(≤),thentheproblemisreferredtoastheSetPackingproblem.Inpractice,theobjectiveoftheSetPackingproblemistypicallyaprofitmaximizationobjective.ThemathematicalmodelunderlyingtheSetPartitioningproblemcanbedescribedasfollows:Assumemdifferentelementsandndifferentsubsetsofthesemelements.Eachsubsetcontainsoneormoreelements.Ifaij=1,thenelementiispartofsubsetj,andifaij=0,thenelementiisnotpartofsubsetj.Theobjectiveistofindacollectionofsubsetssuchthateachelementispartofexactlyonesubset.Theobjectiveistofindthatcollectionofsubsetsthathaveaminimumcost.IntheSetCoveringproblem,eachelementhastobepartofatleastonesubset.IntheSetPackingproblemeachsubsetyieldsacertainprofitπjandthetotalprofithastobemaximizedinsuchawaythateachelementispartofatmostonesubset.AnexampleoftheSetPartitioningproblemisthecrewschedulingproblemconsideredinChapter13.Eachelementisaflightlegandeachsubsetisaroundtrip.Theobjectiveistocovereachflightlegexactlyonceandminimizethetotalcostofalltheroundtrips.EachcolumnintheAmatrixisaroundtripandeachrowisaflightlegthatmustbecoveredexactlyoncebyoneroundtrip.Ifflightlegiispartofroundtripj,thenaijis1,otherwiseaijis0.Letxjdenotea0−1decisionvariablethatassumesthevalue1ifroundtripjisselectedand0otherwise.Theoptimizationproblemisthento
A.6DisjunctiveProgrammingFormulations419select,atminimumcost,asetofroundtripsthatsatisfiestheconstraints.Theconstraintsinthisproblemareoftencalledthepartitioningequations.Forafeasiblesolution(x1,…,xn),thevariablesthatareequalto1arereferredtoasthepartition.AnotherexampleofaSetPartitioningproblemistheaircraftroutingandschedulingproblemconsideredinChapter11.Itisclearthatintheaviationindustrytheconstraintthateachflightlegshouldbecoveredexactlyoncebyaroundtripisimportant.However,inthetruckingindustry,itmaybepossibletohaveonelegcoveredbyseveralroundtrips.TheconstraintsmaythereforeberelaxedandtheproblemthenassumesaSetCoveringstructure.AnexampleofaSetPackingproblemisthetankerschedulingproblemdescribedinChapter11.Thereasonwhythetankerschedulingproblemisapackingproblemandnotapartitioningproblemisbasedonthefactthatnoteverycargohastobetransportedbyacompanyownedtanker.Ifitisadvantageoustoassignacargotoanoutsidecharter,thenthatisallowed.A.6DisjunctiveProgrammingFormulationsThereisalargeclassofmathematicalprogramsinwhichtheconstraintscanbedividedintoasetofconjunctiveconstraintsandoneormoresetsofdisjunctiveconstraints.Asetofconstraintsiscalledconjunctiveifeachoneoftheconstraintshastobesatisfied.Asetofconstraintsiscalleddisjunctiveifatleastoneoftheconstraintshastobesatisfiedbutnotnecessarilyall.Inthestandardlinearprogramallconstraintsareconjunctive.ThemixedintegerprogramdescribedinExampleA.4.1inessencecontainspairsofdis-junctiveconstraints.Thefactthattheintegervariablexjkhastobeeither0or1canbeenforcedbyapairofdisjunctivelinearconstraints:eitherxjk=0orxjk=1.Thisimpliesthatthesinglemachineproblemwithprecedencecon-straintsandthetotalweightedcompletiontimeobjectivecanbeformulatedasadisjunctiveprogramaswell.ExampleA.6.1(ADisjunctiveProgramforaSingleMachine).Be-foreexpressingthenonpreemptivesinglemachineproblemwiththejobssub-jecttoprecedenceconstraintsandthetotalweightedcompletiontimeobjec-tiveintheformofadisjunctiveprogram,itisofinteresttorepresenttheproblembyadisjunctivegraphmodel.LetNdenotethesetofnodeswhichcorrespondtothenjobs.Betweenanypairofnodes(jobs)jandkinthisgraphexactlyoneofthefollowingthreeconditionshastohold:(i)jobjprecedesjobk,(ii)jobkprecedesjobjand(iii)jobsjandkareindependentwithrespecttooneanother.ThesetofdirectedarcsArepresenttheprecedencerelationshipsbetweenthejobs.Thesearcsaretheso-calledconjunctivearcs.LetsetIcontainallthepairsofjobsthatareindependentofoneanother.Eachpairofjobs(j,k)∈I
420AMathematicalProgramming:FormulationsandApplicationsarenowconnectedwithoneanotherbytwoarcsgoinginoppositedirections.Thesearcsarereferredtoasdisjunctivearcs.Theproblemistoselectfromeachpairofdisjunctivearcsbetweentwoindependentjobsjandkonearcwhichindicateswhichoneofthetwojobsgoesfirst.Theselectionofdis-junctivearcshastobesuchthatthesearcstogetherwiththeconjunctivearcsdonotcontainacycle.Theselecteddisjunctivearcstogetherwiththeconjunctivearcsdetermineascheduleforthenjobs.Letthevariablexjinthedisjunctiveprogramformulationdenotethecompletiontimeofjobj.ThesetAdenotesthesetofprecedenceconstraintsj→kwhichrequirejobjtobeprocessedbeforejobk.minimizenj=1wjxjsubjecttoxk−xj≥pkforallj→k∈A,xj≥pjforj=1,…,n,xk−xj≥pkorxj−xk≥pjforall(j,k)∈I.Thefirstandsecondsetofconstraintsareconjunctiveconstraints.Thethirdsetisasetofdisjunctiveconstraints.ExampleA.6.2(ADisjunctiveProgramforaJobShop).TopresentadisjunctiveprogramforajobshopasdescribedinChapter5,letthevariableyijdenotethestartingtimeofoperation(i,j).RecallthatsetNdenotesthesetofalloperations(i,j),andsetAthesetofallroutingconstraints(i,j)→(k,j)whichrequirejobjtobeprocessedonmachineibeforeitisprocessedonmachinek.Thefollowingmathematicalprogramminimizesthemakespan.minimizeCmaxsubjecttoykj−yij≥pijforall(i,j)→(k,j)∈ACmax−yij≥pijforall(i,j)∈Nyij−yil≥piloryil−yij≥pijforall(i,l)and(i,j),i=1,…,myij≥0forall(i,j)∈NInthisformulation,thefirstsetofconstraintsensurethatoperation(k,j)cannotstartbeforeoperation(i,j)iscompleted.Thethirdsetofconstraintsarethedisjunctiveconstraints;theyensurethatsomeorderingexistsamongoperationsofdifferentjobsthathavetobeprocessedonthesamemachine.Becauseoftheseconstraintsthisformulationisreferredtoasadisjunctiveprogrammingformulation.
A.6DisjunctiveProgrammingFormulations421ExampleA.6.3(ADisjunctiveProgramforaJobShopContinued).Considerthefollowingexamplewithfourmachinesandthreejobs.Theroute,i.e.,themachinesequence,aswellastheprocessingtimesaregiveninthetablebelow.jobsmachinesequenceprocessingtimes11,2,3p11=10,p21=8,p31=422,1,4,3p22=8,p12=3,p42=5,p32=631,2,4p13=4,p23=7,p43=3TheobjectiveconsistsofthesinglevariableCmax.Thefirstsetofconstraintsconsistsofsevenconstraints:twoforjob1,threeforjob2andtwoforjob3.Forexample,oneoftheseisy21−y11≥10(=p11).Thesecondsetconsistsoftenconstraints,oneforeachoperation.AnexampleisCmax−y11≥10(=p11).Thesetofdisjunctiveconstraintscontainseightconstraints:threeeachformachines1and2andoneeachformachines3and4(therearethreeoperationstobeperformedonmachines1and2andtwooperationsonmachines3and4).Anexampleofadisjunctiveconstraintisy11−y12≥3(=p12)ory12−y11≥10(=p11).Thelastsetincludestennonnegativityconstraints,oneforeachstartingtime.Thataschedulingproblemcanbeformulatedasadisjunctiveprogramdoesnotimplythatthereisastandardsolutionprocedureavailablethatwillworksatisfactorily.Minimizingthemakespaninajobshopisaveryhardprob-lemandsolutionproceduresareeitherbasedonenumerationoronheuristics.Toobtainoptimalsolutionsbranch-and-boundmethodsarerequired.Thesametechniquesthatcanbeappliedtointegerprogramscanalsobeappliedtodisjunctiveprograms.Theapplicationofbranch-and-boundtoadisjunctiveprogramisstraightforward.FirsttheLPrelaxationofthedisjunc-tiveprogramhastobesolved(i.e.,theLPobtainedafterdeletingthesetofdisjunctiveconstraints).IftheoptimalsolutionoftheLPbychancesatisfiesalldisjunctiveconstraints,thenthesolutionisoptimalforthedisjunctivepro-gramaswell.However,ifoneofthedisjunctiveconstraintsisviolated,saytheconstraintxk−xj≥pkorxj−xk≥pj,thentwoadditionalLP’saregenerated.Onehastheextraconstraintxk−xj≥pkandtheotherhastheextraconstraintxj−xk≥pj.Theprocedureisinallotherrespectssimilartoastandardbranch-and-boundprocedureforintegerprogrammingasdescribedinAppendixB.
422AMathematicalProgramming:FormulationsandApplicationsExercisesA.1.ConsidertheproblemdescribedinExampleA.4.1.Inordertoformulateadifferentintegerprogramforthesameproblemintroducethe0-1decisionvariablexjt.Thisvariableis1ifjobjstartsexactlyattimetand0otherwise.(a)Showthattheobjectiveoftheintegerprogramisminimizenj=1lt=0wj(t+pj)xjt(b)Formulatetheconstraintsetsforthisintegerprogram.A.2.FormulatetheinstanceofExampleA.6.3asanintegerprogramofthetypedescribedinExampleA.4.2.CommentsandReferencesManybookshavebeenwrittenonlinearprogramming,integerprogrammingandcombinatorialoptimization.ExamplesofsomerelativelyrecentonesarePapadim-itriouandSteiglitz(1982),ParkerandRardin(1988),Padberg(1995),Wolsey(1998),NemhauserandWolsey(1999),andSchrijver(2003).Forexcellentsurveysofmathematicalprogrammingformulationsofmachineschedulingproblems,seeBlazewicz,DrorandWeglarz(1991)andVanDeVelde(1991).
AppendixBExactOptimizationMethodsB.1Introduction……………………………423B.2DynamicProgramming…………………..424B.3OptimizationMethodsforIntegerPrograms….428B.4ExamplesofBranch-and-BoundApplications…430B.1IntroductionThereareplanningandschedulingproblemsthatareinherentlyeasy;theycanbeformulatedaslinearprogramsthatarereadilysolvablethroughtheuseofexistingefficientalgorithms.Othereasyproblemscanbesolvedbydif-ferentalgorithmsthatarealsoefficient.Theseefficientalgorithmsareusuallyreferredtoaspolynomialtimealgorithms.Thataproblemcanbesolvedbyanefficient,polynomialtime,algorithmimpliesthatverylargeinstancesofthatproblem,withhundredsoreventhousandsofjobs,stillcanbesolvedinarelativelyshorttimeonacomputer.However,therearemanymoreplanningandschedulingproblemsthatareintrinsicallyveryhard.TheseproblemsarereferredtoasNP-hard.Theyaretypicallycombinatorialproblemsthatcannotbeformulatedaslinearpro-gramsandtherearenosimplerulesoralgorithmsthatyieldoptimalsolutionsinalimitedamountofcomputertime.Often,itmaybepossibletohaveniceandelegantintegerprogrammingordisjunctiveprogrammingformulationsforsuchproblems,butstillsolvingthesetooptimalitymayrequireanenormousamountofcomputertime.TherearevariousclassesofmethodsthatareusefulforobtainingoptimalsolutionsforsuchNP-Hardproblems.OneclassofmethodsisreferredtoasDynamicProgramming.Dynamicprogrammingisoneofthemorewidelyusedtechniquesfordealingwithcombinatorialoptimizationproblems.Itis
424BExactOptimizationMethodsaprocedurethatisbasedonadivideandconquerapproach.DynamicPro-grammingcanbeappliedtoproblemsthataresolvableinpolynomialtime,aswellasproblemsthatareNP-Hard.IfaplanningandschedulingproblemcanbeformulatedasanIntegerPro-gram,thenvariousothertechniquescanbeapplied.ThebestknownmethodsforsolvingIntegerProgramsare:(i)branch-and-boundmethods,(ii)cuttingplane(polyhedral)methods,(iii)hybridmethods.Thefirstclassofmethods,branch-and-bound,isoneofthemostpopularclassoftechniquesusedforIntegerProgramming.Thebranchingreferstoapartitioningofthesolutionspace;eachpartofthesolutionspaceisthencon-sideredseparately.Theboundingreferstothedevelopmentoflowerboundsforpartsofthesolutionspace(assumingtheobjectivehastobeminimized).Ifalowerboundontheobjectiveinonepartofthesolutionspaceislargerthananintegersolutionalreadyfoundinadifferentpartofthesolutionspace,thecorrespondingpartoftheformersolutionspacecanbedisre-garded.Thesecondclassofmethods,cuttingplanemethods,focusesonthelinearprogramrelaxationoftheintegerprogram.Thesemethodsgenerateadditionallinearconstraints,i.e.,cuttingplanes,thathavetobesatisfiedforthevariablestobeinteger.Theseadditionalinequalitiesconstrainthefeasiblesetmorethantheoriginalsetoflinearinequalitieswithoutcuttingoffintegersolutions.SolvingtheLPrelaxationoftheIPwiththeadditionalconstraintsthenyieldsanewsolution,thatmaybeinteger.Ifthesolutionisinteger,theprocedurestopsasthesolutionobtainedisoptimalfortheoriginalIP.Ifthevariablesarenotinteger,moreinequalitiesaregenerated.Hybridmethodstypicallycombineideasfromvariousdifferentapproaches.Forexample,thecuttingplanemethodhasbecomepopularinrecentyearsthroughitsuseincombinationwithbranch-and-bound.Whenbranch-and-boundisusedinconjunctionwithcuttingplanetechniquesitisreferredtoasbranch-and-cut.B.2DynamicProgrammingDynamicprogrammingisbasicallyacompleteenumerationschemethatat-tempts,viaadivideandconquerapproach,tominimizetheamountofcom-putationtobedone.Theapproachsolvesaseriesofsubproblemsuntilitfindsasolutionfortheoriginalproblem.Itdeterminestheoptimalsolutionforeachsubproblemanditscontributiontotheobjectivefunction.Ateachiterationitdeterminestheoptimalsolutionforasubproblem,whichislargerthanallpreviouslysolvedsubproblems.Itfindsasolutionforthecurrentsubprob-lembyutilizingalltheinformationobtainedearlierinthesolutionsofalltheprevioussubproblems.
B.2DynamicProgramming425Dynamicprogrammingischaracterizedbythreetypesofequations,namely(i)initialconditions;(ii)arecursiverelationand(iii)anoptimalvaluefunction.Inschedulingachoicecanbemadebetweenforwarddynamicprogrammingandbackwarddynamicprogramming.Thefollowingexampleillustratestheuseofforwarddynamicprogramming.ExampleB.2.1(ForwardDynamicProgrammingFormulation).Con-siderasinglemachineandnjobs.IfjobjiscompletedattimeCj,thenacosthj(Cj)isincurred.Theproblemistosequencethejobsinsuchawaythattheobjectivenj=1hj(Cj)isminimized.Thisproblemisaveryimportantprobleminschedulingtheoryasitincludesmanyimportantobjectivefunctionsasspecialcases.Theobjec-tiveis,forexample,ageneralizationofthewjTjobjective;theproblemisthereforeNP-hard.LetJdenoteasubsetofthenjobsandassumethatallthejobsinsetJareprocessedbeforeanyoneofthejobsnotinsetJ.LetV(J)=j∈Jhj(Cj),providedthesetofjobsJisprocessedfirst.Thedynamicprogammingfor-mulationoftheproblemisbasedonthefollowinginitialconditions,recursiverelationandoptimalvaluefunction.InitialConditions:V({j})=hj(pj),j=1,…,nRecursiveRelation:V(J)=minj∈JV(J−{j})+hj(k∈Jpk)OptimalValueFunction:V({1,…,n})Theideabehindthisdynamicprogrammingprocedureisrelativelystraight-forward.Ateachiterationtheoptimalsequenceforasubsetofthejobs(sayasubsetJthatcontainsljobs)isdetermined,assumingthissubsetgoesfirst.Thisisdoneforeverysubsetofsizel.Therearen!/(l!(n−l)!)suchsubsets.Foreachsubsetthecontributionofthelscheduledjobstotheobjectivefunc-tioniscomputed.Throughtherecursiverelationthisisexpandedtoevery
426BExactOptimizationMethodssubsetwhichcontainsl+1jobs.Eachoneofthel+1jobsisconsideredasacandidatetogofirst.Whenusingtherecursiverelationtheactualsequenceoftheljobsofthesmallersubsetdoesnothavetobetakenintoconsideration;onlythecontributionoftheljobstotheobjectivehastobeknown.AfterthevalueV({1,…,n})hasbeendeterminedtheoptimalsequenceisobtainedthroughasimplebacktrackingprocedure.Thecomputationalcomplexityofthisproblemcanbedeterminedasfol-lows.ThevalueofV(J)hastobedeterminedforallsubsetsthatcontainljobs.Therearen!/l!(n−l)!subsets.Sothetotalnumberofevaluationsthathavetobedonearenl=1n!l!(n−l)!=O(2n).ExampleB.2.2(ApplicationofForwardDynamicProgramming).Considertheproblemdescribedinthepreviousexamplewiththefollowingjobs.jobs123pj436hj(Cj)C1+C213+C328C3SoV({1})=20,V({2})=30andV({3})=48.Theseconditerationoftheprocedureconsidersallsetscontainingtwojobs.ApplyingtherecursiverelationyieldsV({1,2})=minV({1})+h2(p1+p2),V({2})+h1(p2+p1)=min(20+346,30+56)=86Soifjobs1and2precedejob3,thenjob2hastogofirstandjob1hastogosecond.InthesamewayitcanbedeterminedthatV({1,3})=100withjob1goingfirstandjob3goingsecondandthatV({2,3})=102withjob2goingfirstandjob3goingsecond.Thelastiterationoftheprocedureconsidersset{1,2,3}.V({1,2,3})=minV({1,2})+h3(p1+p2+p3),V({2,3})+h1(p1+p2+p3),V({1,3})+h2(p1+p2+p3).SoV({1,2,3})=min86+104,102+182,100+2200=190.Itfollowsthatjobs1and2havetogofirstandjob3last.Theoptimalsequenceis2,1,3withobjectivevalue190.
B.2DynamicProgramming427Inthefollowingexamplethesameproblemishandledthroughabackwarddynamicprogrammingprocedure.Inschedulingproblemsthebackwardver-siontypicallycanbeusedonlyforproblemswithamakespanthatissched-uleindependent(e.g.,singlemachineproblemswithoutsequencedependentsetups,multiplemachineproblemswithjobsthathaveidenticalprocessingtimes).Theuseofbackwardsdynamicprogrammingisneverthelessimportantasitissomewhatsimilartothedynamicprogrammingprocedurediscussedinthenextsectionforstochasticschedulingproblems.ExampleB.2.3(BackwardDynamicProgrammingFormulation).Consideragainasinglemachineschedulingproblemwithhj(Cj)asob-jectiveandnopreemptions.ItisclearthatthemakespanCmaxisscheduleindependentandthatthelastjobiscompletedatCmaxwhichisequaltothesumofthenprocessingtimes.Again,JdenotesasubsetofthenjobsanditisassumedthatJisprocessedfirst.LetJCdenotethecomplementofJ.SosetJCisprocessedlast.LetV(J)denotetheminimumcontributionofthesetJCtotheobjectivefunction.Inotherwords,V(J)representtheminimumadditionalcosttocompleteallremainingjobsafteralljobsinsetJhavebeencompleted.Thebackwarddynamicprogrammingprocedureisnowcharacterizedbythefollowinginitialconditions,recursiverelationandoptimalvaluefunction.InitialConditions:V({1,…,j−1,j+1,…,n})=hj(Cmax)j=1,…,nRecursiveRelation:V(J)=minj∈JCV(J∪{j})+hj(k∈J∪{j}pk)OptimalValueFunction:V(∅)Again,theprocedureisrelativelystraightforward.Ateachiteration,theopti-malsequenceforasubsetofthenjobs,sayasubsetJCofsizel,isdetermined,assumingthissubsetgoeslast.Thisisdoneforeverysubsetofsizel.Throughtherecursiverelationthisisexpandedforeverysubsetofsizel+1.Theopti-malsequenceisobtainedwhenthesubsetcomprisesalljobs.Notethat,asinExampleB.2.1,subsetJgoesfirst;however,inExampleB.2.1setJdenotesthesetofjobsalreadyscheduledwhileinthisexamplesetJdenotesthesetofjobsstilltobescheduled.
428BExactOptimizationMethodsB.3OptimizationMethodsforIntegerProgramsIntegerprogramsareoftensolvedviabranch-and-bound.Averybasicbranch-and-boundprocedurecanbedescribedasfollows.SupposeonesolvestheLPrelaxationofanIP(thatis,theIPwithouttheintegralityconstraints).IfthesolutionoftheLPrelaxationhappenstobeinteger,say¯x0,thenthissolutionisoptimalfortheoriginalintegerprogramaswell.If¯x0isnotinteger,thenthevalueoftheoptimalsolutionoftheLPrelaxation,¯c¯x0,stillservesasalowerboundforthevalueoftheoptimalsolutionfortheoriginalintegerprogram.Ifoneofthevariablesin¯x0,isnotinteger,sayxj=r,thenthebranch-and-boundprocedureproceedsasfollows.Theintegerprogrammingproblemissplitintotwosubproblemsbyaddingtwomutuallyexclusiveandexhaustiveconstraints.Inonesubproblem,saySP(1),theoriginalintegerprogramismodifiedbyaddingtheadditionalconstraintxj≤r,whererdenotesthelargestintegersmallerthanr,whileintheothersub-problem,saySP(2),theoriginalintegerprogramismodifiedbyaddingtheadditionalconstraintxj≥rwhererdenotesthesmallestintegerlargerthanr.Itisclearthattheoptimalsolutionoftheoriginalintegerprogramhastolieinthefeasibleregionofoneofthesetwosubproblems.Thebranch-and-boundprocedurenowconsiderstheLPrelaxationofoneofthesubproblems,saySP(1),andsolvesit.Ifthesolutionisinteger,thenthisbranchofthetreedoesnothavetobeexploredfurther,asthissolutionistheoptimalsolutionoftheoriginalintegerprogrammingversionofSP(1).Ifthesolutionisnotinteger,SP(1)hastobesplitintotwosubproblems,saySP(1,1)andSP(1,2)throughtheadditionofmutuallyexclusiveandexhaus-tiveconstraints.Proceedinginthismanneratreeisconstructed.Fromeverynodethatcorrespondstoanonintegersolutionabranchingoccurstotwoothernodes,andsoon.Theboundingprocessisstraightforward.Ifasolutionatanodeisnoninteger,thenthisvalueprovidesalowerboundforallthesolutionsinitsoffspring.Thebranch-and-boundprocedurestopswhenallnodesofthetreeeitherhaveanintegersolutionoranonintegersolutionthatishigherthananintegersolutionatanothernode.Thenodewiththebestintegersolutionprovidesanoptimalsolutionfortheoriginalintegerprogram.Anenormousamountofresearchandexperimentationhasbeendoneonbranch-and-boundtechniques.Forexample,thebranchingtechniqueaswellastheboundingtechniquedescribedabovearerelativelysimple.Severalmoresophisticatedwaysofapplyingbranch-and-boundhaveproventobeveryuse-fulinpractice,namely:(i)Lagrangeanrelaxation,
B.3OptimizationMethodsforIntegerPrograms429(ii)branch-and-cut,and(iii)branch-and-price(oftenalsoreferredtoascolumngeneration).Lagrangeanrelaxationisasophisticatedtechniqueforestablishinglowerboundsinabranch-and-boundprocedure.Itgenerateslowerboundsthataresubstantiallybetter(higher)thantheLPrelaxationboundsdescribedaboveandbetterboundstypicallycutdowntheoverallcomputationtimesubstan-tially.Lagrangeanrelaxation,insteadofdroppingtheintegralityconstraints,relaxessomeofthemainconstraints.However,therelaxedconstraintsarenottotallydropped.Instead,theyaredualizedorweightedintheobjectivefunctionwithsuitableLagrangeanmultiplierstodiscourageviolations.Branch-and-cutisaclassofmethodsthatarebasedonacombinationofbranch-and-boundwithcuttingplanetechniques.Branch-and-cutusesineachsubproblemofthebranchingtreeacuttingplanealgorithmtogeneratealowerbound.Thatis,acuttingplanealgorithmisappliedtotheprob-lemformulationthatincludestheadditionalconstraintsintroducedatthatnode.Abranch-and-cutmethodsolvesasequenceoflinearprogrammingrelax-ationsoftheintegerprogrammingproblemtobesolved;thecuttingplanemethodimprovestherelaxationoftheproblemtomorecloselyapproximatetheintegerprogrammingproblem,andthebranch-and-boundproceedsbytheusualdivideandconquerapproachtosolvetheproblem.Apurebranch-and-boundapproachcanbespedupconsiderablybytheemploymentofacuttingplanescheme,eitherjustatthetopofthetree,orateverynodeofthetree.Lately,thesetechniqueshavefoundapplicationsincrewschedulingandtruckdispatchingproblems.Branch-and-priceisoftenusedtosolveintegerprogramsthathaveahugenumberofvariables(columns).Abranch-and-pricealgorithmalwaysworkswitharestrictedprobleminasensethatonlyasubsetofthevariablesistakenintoaccount;thevariablesoutsidethesubsetarefixedat0andthecorrespondingcolumnsaredisregarded.FromthetheoryofLinearProgram-mingitisknownthataftersolvingthisrestrictedproblemtooptimality,eachvariablethatisincludedhasanegativepotentialsavingsor,equivalently,apositiveso-calledreducedcost.Ifeachvariablethatisnotincludedintherestrictedproblemalsohasanegativepotentialsavings,thenanoptimalso-lutionfortheoriginalproblemisfound.However,iftherearevariableswithapositivepotentialsavings,thenoneormoreofthesevariablesshouldbein-cludedintherestrictedproblem.Themainideabehindcolumngenerationisthattheoccurrenceofvariableswithpositivepotentialsavingsisnotverifiedbyenumeratingallvariables,butratherbysolvinganoptimizationproblem.Thisoptimizationproblemiscalledthepricingproblemandisdefinedastheproblemoffindingthevariablewithmaximumpotentialsavings(orminimumreducedcost).Toapplycolumngenerationeffectivelyitisimportanttofindagoodmethodforsolvingthepricingproblem.Abranch-and-boundalgorithminwhichthelowerboundsarecomputedbysolvingLPrelaxationsthroughcolumngenerationiscalledabranch-cut-and-pricealgorithm.Branch-and-
430BExactOptimizationMethodspriceandbranch-cut-and-pricehavebeenappliedsuccessfullytovariouspar-allelmachineschedulingproblems.B.4ExamplesofBranch-and-BoundApplicationsInthissectionweillustratetheuseofbranch-and-boundwithtwoexamples.First,considerasinglemachineandnjobs.Jobjhasreleasedaterjandduedatedj.Theobjectiveistominimizethemaximumlatenesswithoutpreemptions.ThisproblemisNP-hard.Itisimportantbecauseitappearsfrequentlyasasubprobleminheuristicproceduresforflowshopandjobshopscheduling(see,forexample,theshiftingbottleneckproceduredescribedinChapter5).Ithasreceivedthereforeconsiderableattention,thathasresultedinanumberofreasonablyefficientbranch-and-boundprocedures.Abranch-and-boundprocedureforthisproblemcanbedesignedasfollows.Thebranchingprocessmaybebasedonthefactthatschedulesaredevelopedstartingfromthebeginningoftheschedule.Thereisasinglenodeatlevel0whichisatthetopofthetree.Atthisnodenoneofthejobshavebeenputintoanypositioninthesequence.Therearenbranchesgoingdowntonnodesatlevel1.Eachnodeatthislevelcorrespondstoapartialsolutionwithaspecificjobinthefirstpositionoftheschedule.So,ateachofthesenodestherearestilln−1jobswhosepositionsinthescheduleareyettobedetermined.Therearen−1arcsemanatingfromeachnodeatlevel1tolevel2.Therearethereforen×(n−1)nodesatlevel2.Ateachnodeatlevel2,thetwojobsinthefirsttwopositionsarespecified;atlevelk,thejobsinthefirstkpositionsarefixed.Actually,itisoftennotnecessarytoconsidereveryremainingjobasacandidateforthenextposition.Ifatanodeatlevelk−1jobsj1,…,jk−1areassignedtothefirstk−1positions,jobchastobeconsideredasacandidateforpositionkonlyifrc0arecontrolparametersreferredtoascoolingparametersortemperatures(inanalogywiththeannealingprocessmentionedabove).Oftenβkischosentobeakforsomeabetween0and1.
456CHeuristicMethodsFromtheabovedescriptionofthesimulatedannealingprocedureitisclearthatmovestoworsesolutionsareallowed.Thereasonforallowingthesemovesistogivetheproceduretheopportunitytomoveawayfromalocalminimumandfindabettersolutionlateron.Sinceβkdecreaseswithk,theacceptanceprobabilityforanon-improvingmoveislowerinlateriterationsofthesearchprocess.Thedefinitionoftheacceptanceprobabilityalsoensuresthatifaneighborissignificantlyworse,itsacceptanceprobabilityisverylowandthemoveisunlikelytobemade.Severalstoppingcriteriaareusedforthisprocedure.Onewayistolettheprocedurerunforaprespecifiednumberofiterations.Anotheristolettheprocedurerununtilnoimprovementhasbeenobtainedforagivennumberofiterations.Themethodcanbesummarizedasfollows:AlgorithmC.5.2(SimulatedAnnealing).Step1.Setk=1andselectβ1.SelectaninitialsequenceS1usingsomeheuristic.SetS0=S1.Step2.SelectacandidatescheduleScfromtheneighbourhoodofSk.IfG(S0)G(Sk)generatearandomnumberUkfromaUniform(0,1)distribution;IfUk≤P(Sk,Sc)setSk+1=ScotherwisesetSk+1=SkandgotoStep3.Step3.Selectβk+1≤βk.Incrementkby1.Ifk=NthenSTOP,otherwisegotoStep2.Theeffectivenessofsimulatedannealingdependsonthedesignoftheneighbourhoodandonhowthesearchisconductedwithinthisneighbourhood.Iftheneighbourhoodisdesignedinawaythatfacilitatesmovestobettersolutionsandmovesoutoflocalminima,thentheprocedurewillperformwell.Thesearchwithinaneighbourhoodcanbedonerandomlyorinamoreorganizedway.Forexample,thecontributionofeachjobtotheobjectivefunctioncanbecomputedandthejobwiththehighestimpactontheobjectivecanbeselectedasacandidateforaninterchange.
C.5LocalSearch:SimulatedAnnealingandTabu-Search457Overthelasttwodecadessimulatedannealinghasbeenappliedtomanyschedulingproblems,inacademiaaswellasinindustry,withconsiderablesuccess.Intheremainingpartofthissectionwedescribethetabu-searchproce-dure.Tabu-searchisinmanywayssimilartosimulatedannealinginthatitalsomovesfromonescheduletoanotherwiththenextschedulebeingpossiblyworsethantheonebefore.Foreachschedule,aneighbourhoodisdefinedasinsimulatedannealing.Thesearchforaneighborwithintheneighbourhoodasapotentialcandidatetomovetoisagainadesignissue.Asinsimulatedannealing,thiscanbedonerandomlyorinanorganizedway.Thebasicdif-ferencebetweentabu-searchandsimulatedannealingliesinthemechanismusedforapprovingacandidateschedule.Intabu-searchthemechanismisnotprobabilisticbutratherofadeterministicnature.Atanystageoftheprocessatabu-listofmutations,whichtheprocedureisnotallowedtomake,iskept.Mutationsonthetabu-listmaybe,forexample,pairsofjobsthatmaynotbeinterchanged.Thetabu-listhasafixednumberofentries(usuallybetween5and9),whichdependsupontheapplication.Everytimeamoveismadebyamutationinthecurrentschedule,thereversemutationisenteredatthetopofthetabu-list;allotherentriesarepusheddownonepositionandthebottomentryisdeleted.Thereversemutationisputonthetabu-listtoavoidreturningtoalocalminimumthathasbeenvisitedbefore.Actually,attimesareversemutationthatistabucouldhaveledtoanewschedule,notvisitedbefore,withanobjectivevaluelowerthananyoneobtainedbefore.Thismayhappenwhenthemutationisclosetothebottomofthetabu-listandanum-berofmoveshavealreadybeenmadesincethemutationwasputonthelist.Thus,ifthenumberofentriesinthetabu-lististoosmallcyclingmayoccur;ifitistoolargethesearchmaybeundulyconstrained.Themethodcanbesummarizedasfollows:AlgorithmC.5.3(Tabu-Search).Step1.Setk=1.SelectaninitialsequenceS1usingsomeheuristic.SetS0=S1Step2.SelectacandidatescheduleScfromtheneighbourhoodofSk.IfthemoveSk→Scisprohibitedbyamutationonthetabu-list,setSk+1=SkandgotoStep3.IfthemoveSk→Scisnotprohibitedbyanymutationonthetabu-list,setSk+1=Scandenterreversemutationatthetopofthetabu-list;pushallotherentriesinthetabu-listonepositiondown;deletetheentryatthebottomofthetabu-list.
458CHeuristicMethodsIfG(Sc)color[whichcar[j+1]]);18.penaltycost=sum(jincarrng)19.(maxl(whencar[j]-position[j],0)+(maxl(position[j]-(whencar[j]+OKinterval),0));20.alldifferent(position);21.alldifferent(whichcar);22.forall(jincarrng){23.whichcar[position[j]]=j;24.position[whichcar[j]]=j;25.}26.forall(cinColors){27.forall(kinwhencolor[c]:k<>whencolor[c].last()){28.position[k]#include#include#include#include//Weneedadatastructuretostoreinformationaboutjob.structTjob{intid;intrelease;intdue;intweight;intproc;doublewp;//weightdividedbyprocessingtime};TjobjobArray[100];intjobCount;//astringbuffer.charbuffer[1024];//———————————————————-//Forthesinglemachinesetting,wedonothavetoread//themachinefile.Butwewilldoithereanyhowjustto//verifythatitactuallyrepresentsasinglemachine.voidReadMch(){//Noneedtousethequalifiedpath.Just”user.mch”.ifstreamFmch(“user.mch”,ios::nocreate);//Checkthefirstlineinthefile.//Ifitisnot”Single:”,toobad.Fmch.getline(buffer,1024);if(strcmp(buffer,”Single:”)){cout<<"wedonotsupportflexibleworkstations!\n";exit(1);}//Nowweskipseverallines.Therearetwowaystoskip://Getlineorignore.Getlineallowsyoutocheckwhatyouareskipping.Fmch.getline(buffer,1024);//buffer=”Workstation:Wks000”,//butwedonotcare.Fmch.ignore(1024,’\n’);//skip”Setup:”Fmch.ignore(1024,’\n’);//skip”Machine:”//Wedonotneedtheavailabilitytimeorthestartingstatusforthe F.2LinkingExternalAlgorithms483//machine,butwewillreaditjusttoshowhowitisdone.Fmch.ignore(20);//skip”Release:”intavail:Fmch>>avail;Fmch.ignore(20)//skip”Status:”//Countingspacesisnotagoodidea,sojustreadtillthefirstcharacter.Fmch.eatwhite();charstatus=Fmch.get();//Nowtherestofthefilemustcontainonlywhite-spacecharacters.Fmch.eatwhite();if(!Fmch.eof()){cout<<"Thefilemustcontainatleasttwoworkstations!\n";exit(1);}//----------------------------------------------------------//Withthejobfileitislesseasy;astreamofjobshavetoberead.voidreadJob(){ifstreamFjob("user.job",ios::nocreate);Fjob>>buffer;//buffer=”Shop:”,ignoreFjob>>buffer;//checkifsinglemachineif(strcmp(buffer,”Single”)){cout<<"Thisisnotasinglemachinefile!\n";exit(1);}while(1){Fjob>>buffer;}//buffer=”Job:”if(strcmp(buffer,”Job:”))//ifnot,mustbetheendofthefilebreak;Fjob>>buffer;//buffer=”Job###”,ignorejobarray[jobCount].id=jobCount;Fjob>>buffer;//buffer=”release:’Fjob>>jobArray[jobCount].release;Fjob>>buffer;//buffer=”due:’Fjob>>jobArray[jobCount].due;Fjob>>buffer;//buffer=”weight:’Fjob>>jobArray[jobCount].weight;Fjob>>buffer;//buffer=”Oper:’Fjob>>buffer;//buffer=”Wks000;#;A”andweneedthe#char*ss=strchr(buffer,’;’);if(!ss)break;if(sscanf(ss+1,”%d”,&jobArray[jobCount].proc)<1)break;jobArray[jobCount].wp=double(jobArray[jobCount].weight/jobArray[jobCount].proc);jobcount++;}if(jobCount==0) 484FTheLekinSystemUser’sGuide{cout<<"Nojobsdefined!\n";exit(1);}}//----------------------------------------------------------//Comparefunctionforsortingintcompare(constvoid*j1,constvoid*j2){TJob*jb1=(TJob*)j1;TJob*jb2=(TJob*)j2;doublea=jb1->wp-jb2->wp;returna<0?-1:a>0?1:0;}//Sincethisisjustasinglemachine,//wecanimplementanyrulebysortingonthejobarray.//WeusethatCstandardqsortfunction.voidSortJobs(){qsort(jobArray,jobCount,sizeof(TJob),compare);}//Outputtheschedulefile.voidWriteSeq(){ofstreamFsch(“user.seq”);Fsch<<"Schedule:WSPTrule\n";//schedulenameFsch<<"Machine:Wks000.000\n";//Nameoffirstandlastmachine//Nowenumeratetheoperations.for(inti=0;i

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